MATLAB: Attempting to find a fit to this data

analytical functioncurve fitting

Hi there,
Looking to find a curve fit to this data as shown (and attached) in this post. The catch is that it's 3D data, not 2D…
I am aware of the Kriging function, however I require a function where I know the form of the resulting function so that I can analytically find its derivative.
Any thoughts would be appreciated…

Best Answer

>> f=fit([x,y],log(z),'poly55')
     Linear model Poly55:
     f(x,y) = p00 + p10*x + p01*y + p20*x^2 + p11*x*y + p02*y^2 + p30*x^3 + p21*x^2*y 
                    + p12*x*y^2 + p03*y^3 + p40*x^4 + p31*x^3*y + p22*x^2*y^2 
                    + p13*x*y^3 + p04*y^4 + p50*x^5 + p41*x^4*y + p32*x^3*y^2 
                    + p23*x^2*y^3 + p14*x*y^4 + p05*y^5
     Coefficients (with 95% confidence bounds):
       p00 =       21.69  (21.5, 21.89)
       p10 =      0.8578  (-0.2567, 1.972)
       p01 =      0.3813  (-1.914, 2.677)
       p20 =       9.216  (3.857, 14.58)
       p11 =      -12.69  (-18.11, -7.278)
       p02 =      0.1147  (-9.909, 10.14)
       p30 =      -54.22  (-67.48, -40.97)
       p21 =       31.53  (21.3, 41.76)
       p12 =       16.31  (4.092, 28.53)
       p03 =      -1.082  (-21.32, 19.16)
       p40 =       82.65  (67.27, 98.03)
       p31 =       7.202  (-3.904, 18.31)
       p22 =         -55  (-65.89, -44.11)
       p13 =      0.9892  (-12.07, 14.05)
       p04 =      0.2342  (-18.94, 19.41)
       p50 =       -38.6  (-45.28, -31.92)
       p41 =       -29.1  (-34.41, -23.79)
       p32 =       24.53  (19.51, 29.56)
       p23 =       15.26  (10.15, 20.37)
       p14 =      -4.911  (-10.44, 0.6136)
       p05 =      0.4245  (-6.466, 7.315)
>> zhat=exp(f(x,y));
>> hS=scatter3(x,y,zhat,'r');
>>
isn't too bad it appears--
The log transform was needed to counteract the large range of the interpolant; standardizing it might help further.
But, MUST the derivative be analytic over a single function? Could you not use spline interpolant that is differentiable through the pieces?
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