随机排列的坐标值的3D插值

Question

In a .NET/C# project I have relativley randomly arranged coordinates and assigned a weight KPI to each one of them like this:

Weight   | Lat       | Long
---------+-----------+-----------
1,123444 | 54,219367 | 9,696117
1,067940 | 51,950265 | 11,692274
1,054050 | 52,131392 | 13,216249
1,044820 | 52,636704 | 9,845077
1,023757 | 50,118346 | 7,308953
1,006427 | 53,612651 | 12,429595
1,002610 | 51,433237 | 7,661594
0,988619 | 49,396423 | 7,022961
0,955858 | 51,010989 | 10,845346
0,943669 | 50,652052 | 9,162438
0,938015 | 51,104541 | 13,201738
0,863415 | 48,790447 | 11,497889
0,835913 | 48,661604 | 9,350134

I am not an expert when it comes to mathematics or statistics. What I need to do is to interpolate any other given geo-coordinates weight based on the previously calculated ones (see above).

All the 3D-interpolation algorithms I found assume that the coordinates would have to be arranged in kind of a rectangular shape which is not possible in my szenario.

Are there any algorithms which would fit my need? It might be helpful to point out, that we are using a bit of R.NET funktionality in the same project, so a solution using R would be perfectly fine.

Answer 1

There are different methods to do interpolation and they give you slightly different results. You should try a couple and see what works best for your scenario.

One way to do it is take all the data points on a certain radius. Compute the distance to each, then compute a weighted average of their heights using he distance as the weight. This will allow you to interpolate linearly between the points.

If you want something a bit more wavy you can do the same but instead of using distance as weight, you use distance^2 or distance^3. This will give more weight to the closest point and make the surface more curved. You can even combine them to have something like a*distance^2 + b*distance but I would keep it as simple as possible.

If you don't want to use a radius because of uneven distribution, you can take the closest K points (experiment with K and see what works best).

These are all very generic but they work relatively well in a lot of cases.

Note: you can compute distance in 3D space if the points are relatively close, but if they are far away you should compute the distance over the sphere (the length of the arc).