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MathGroup Archive 1995

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Re: suggestions?

  • To: mathgroup at christensen.cybernetics.net
  • Subject: [mg554] Re: [mg549] suggestions?
  • From: el at qua.crl.melco.co.jp (E. Lange)
  • Date: Thu, 16 Mar 95 14:22:53 JST

> There is a rectangular plot of land, several acres in size, which is
> contaminated by a single hazardous waste. We have some chemical
> measurements of the concentation of the waste in the soils at N randomly
> spaced points -- these points are NOT on a grid.
> 
> From other information about the deposition of the waste, we know that the
> function
> 
>         concentration( x, y )
> 
> is well behaved and smoothly varying.....
> 
> So, I want to use a 2D InterpolationFunction in Mma to model the

You might try a simple interpolation using radial basis functions:

data = Table[Random[], {10}, {3}]
c = 5

Transpose[data][[3]] . # / Plus @@ # & [
        E^(-c^2((x-#[[1]])^2 + (y-#[[2]])^2))& /@ data ]
Plot3D[%, {x, 0, 1}, {y, 0, 1}]

The parameter c adjusts the `stiffness' of the interpolation.
For c = 0, the interpolation result is the average concentration of the
waste. For c-->Infinity, the interpolation result is the concentration
at the closest measurement point. You can also try to use other basis
functions, such as 1/d^c instead of E^(-c^2 d), or incorporate
knowledge about the physics of the problem.

In readable notation:

InterpolationRBF[data_, c_][x_, y_] :=
    Module[{radial},
        radial = Table[ E^(-c^2 (
                    (x - data[[i, 1]])^2 + (y - data[[i, 2]])^2) ),
                    {i, Length[data]} ];
        Transpose[data][[3]] . radial / Apply[Plus, radial]
    ]
f = InterpolationRBF[data, c];
Show[ Plot3D[f[x, y], {x, 0, 1}, {y, 0, 1}],
      Graphics3D[Point /@ data] ]

Eberhard Lange


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