Problem with fit

Question

I am trying to find a proper matching function for the following data set:

xyzValues = Import["nC_Q4.csv", "Data"][[8 ;;, {6, 9, 38}]]

{{0, 0, 2.0004}, {0, 100, 0}, {0, 40, 0}, {0, 140, 0}, {0, 60, 0}, {0,
   80, 0}, {0, 120, 0}, {0, 20, 0}, {0, 160, 0}, {0, 180, 0}, {0, 240,
   0}, {0, 220, 0}, {0, 200, 0}, {0, 260, 0}, {0, 280, 0}, {0, 300, 
  0}, {0, 320, 0}, {0, 340, 0}, {0, 360, 0}, {0, 400, 0}, {0, 380, 
  0}, {0, 440, 0}, {0, 420, 0}, {0, 460, 0}, {5, 0, 1.1607}, {5, 20, 
  0}, {5, 40, 0}, {5, 60, 0}, {5, 80, 0}, {5, 100, 0}, {0, 480, 
  0}, {0, 500, 0}, {5, 120, 0}, {5, 140, 0}, {5, 160, 0}, {5, 180, 
  0}, {5, 200, 0}, {5, 240, 0}, {5, 220, 0}, {5, 260, 0}, {5, 280, 
  0}, {5, 300, 0}, {5, 320, 0}, {5, 340, 0}, {5, 360, 0}, {5, 380, 
  0}, {5, 400, 0}, {5, 420, 0}, {10, 20, 0}, {5, 440, 0}, {10, 0, 
  0.480288}, {5, 460, 0}, {10, 60, 0}, {5, 480, 0}, {5, 500, 0}, {10, 
  40, 0}, {10, 80, 0}, {10, 100, 0}, {10, 120, 0}, {10, 140, 0}, {10, 
  160, 0}, {10, 180, 0}, {10, 200, 0}, {10, 220, 0}, {10, 240, 
  0}, {10, 260, 0}, {10, 280, 0}, {10, 300, 0}, {10, 320, 0}, {10, 
  340, 0}, {10, 360, 0}, {10, 380, 0}, {10, 400, 0}, {10, 420, 
  0}, {10, 460, 0}, {10, 480, 0}, {10, 440, 0}, {15, 0, 0.5003}, {10, 
  500, 0}, {15, 20, 0}, {15, 40, 0}, {15, 60, 0}, {15, 100, 0}, {15, 
  80, 0}, {15, 120, 0}, {15, 140, 0}, {15, 160, 0}, {15, 180, 0}, {15,
   200, 0}, {15, 220, 0}, {15, 240, 0}, {15, 260, 0}, {15, 280, 
  0}, {15, 300, 0}, {15, 320, 0}, {15, 340, 0}, {15, 360, 0}, {15, 
  380, 0}, {15, 400, 0}, {15, 420, 0}, {15, 460, 0}, {15, 440, 
  0}, {15, 480, 0}, {15, 500, 0}, {20, 0, 0.520104}, {20, 20, 0}, {20,
   40, 0}, {20, 60, 0}, {20, 80, 0}, {20, 100, 0}, {20, 120, 0}, {20, 
  140, 0}, {20, 160, 0}, {20, 180, 0}, {20, 200, 0}, {20, 220, 
  0}, {20, 240, 0}, {20, 260, 0}, {20, 280, 0}, {20, 300, 0}, {20, 
  320, 0}, {20, 340, 0}, {20, 360, 0}, {20, 380, 0}, {20, 400, 
  0}, {20, 420, 0}, {20, 440, 0}, {20, 460, 0}, {20, 480, 0}, {20, 
  500, 0}, {25, 0, 0.980392}, {25, 20, 0}, {25, 40, 0}, {25, 80, 
  0}, {25, 60, 0}, {25, 100, 0}, {25, 120, 0}, {25, 140, 0}, {25, 160,
   0}, {25, 180, 0}, {25, 200, 0}, {25, 220, 0}, {25, 240, 0}, {25, 
  260, 0}, {25, 280, 0}, {25, 300, 0}, {25, 320, 0}, {25, 340, 
  0}, {25, 360, 0}, {25, 380, 0}, {25, 400, 0}, {25, 420, 0}, {25, 
  440, 0}, {25, 480, 0}, {25, 460, 0}, {25, 500, 0}, {30, 0, 
  2.36047}, {30, 20, 0}, {30, 40, 0}, {30, 60, 0}, {30, 80, 0}, {30, 
  100, 0}, {30, 120, 0}, {30, 140, 0}, {30, 160, 0}, {30, 180, 
  0}, {30, 200, 0}, {30, 220, 0}, {30, 240, 0}, {30, 260, 0}, {30, 
  280, 0}, {30, 300, 0}, {30, 320, 0}, {30, 340, 0}, {30, 360, 
  0}, {30, 380, 0}, {30, 400, 0}, {30, 420, 0}, {30, 440, 0}, {30, 
  460, 0}, {30, 480, 0}, {30, 500, 0}, {35, 0, 14.943}, {35, 20, 
  0}, {35, 40, 0}, {35, 60, 0}, {35, 80, 0}, {35, 100, 0}, {35, 120, 
  0}, {35, 140, 0}, {35, 160, 0}, {35, 180, 0}, {35, 200, 0}, {35, 
  220, 0}, {35, 240, 0}, {35, 260, 0}, {35, 280, 0}, {35, 300, 
  0}, {35, 320, 0}, {35, 340, 0}, {35, 360, 0}, {35, 380, 0}, {35, 
  400, 0}, {35, 420, 0}, {35, 440, 0}, {35, 460, 0}, {35, 480, 
  0}, {35, 500, 0}, {40, 0, 28.6972}, {40, 20, 0}, {40, 40, 0}, {40, 
  60, 0}, {40, 80, 0}, {40, 100, 0}, {40, 120, 0}, {40, 140, 0}, {40, 
  160, 0}, {40, 180, 0}, {40, 200, 0}, {40, 220, 0}, {40, 240, 
  0}, {40, 260, 0}, {40, 280, 0}, {40, 320, 0}, {40, 300, 0}, {40, 
  340, 0}, {40, 360, 0}, {40, 380, 0}, {40, 400, 0}, {40, 420, 
  0}, {40, 440, 0}, {40, 460, 0}, {40, 480, 0}, {40, 500, 0}, {45, 0, 
  34.7878}, {45, 20, 0}, {45, 40, 0}, {45, 60, 0}, {45, 80, 0}, {45, 
  100, 0}, {45, 120, 0}, {45, 140, 0}, {45, 160, 0}, {45, 180, 
  0}, {45, 200, 0}, {45, 220, 0}, {45, 240, 0}, {45, 260, 0}, {45, 
  280, 0}, {45, 300, 0}, {45, 320, 0}, {45, 340, 0}, {45, 360, 
  0}, {45, 380, 0}, {45, 400, 0}, {45, 420, 0}, {45, 440, 0}, {45, 
  460, 0}, {45, 480, 0}, {45, 500, 0}, {50, 0, 39.7718}, {50, 20, 
  0}, {50, 40, 0}, {50, 60, 0}, {50, 80, 0}, {50, 100, 0}, {50, 120, 
  0}, {50, 140, 0}, {50, 160, 0}, {50, 180, 0}, {50, 200, 0}, {50, 
  220, 0}, {50, 260, 0}, {50, 240, 0}, {50, 280, 0}, {50, 300, 
  0}, {50, 320, 0}, {50, 340, 0}, {50, 360, 0}, {50, 380, 0}, {50, 
  400, 0}, {50, 420, 0}, {50, 440, 0}, {50, 460, 0}, {50, 480, 
  0}, {50, 500, 0}, {55, 0, 47.2884}, {55, 20, 0}, {55, 40, 0}, {55, 
  60, 0}, {55, 80, 0}, {55, 100, 0}, {55, 120, 0}, {55, 140, 0}, {55, 
  160, 0}, {55, 180, 0}, {55, 200, 0}, {55, 220, 0}, {55, 240, 
  0}, {55, 260, 0}, {55, 280, 0}, {55, 300, 0}, {55, 320, 0}, {55, 
  340, 0}, {55, 360, 0}, {55, 400, 0}, {55, 380, 0}, {55, 420, 
  0}, {55, 440, 0}, {55, 460, 0}, {55, 480, 0}, {55, 500, 0}, {60, 0, 
  62.3125}, {60, 20, 0.22022}, {60, 40, 0}, {60, 60, 0}, {60, 80, 
  0}, {60, 100, 0}, {60, 120, 0}, {60, 140, 0}, {60, 160, 0}, {60, 
  180, 0}, {60, 200, 0}, {60, 220, 0}, {60, 240, 0}, {60, 260, 
  0}, {60, 280, 0}, {60, 300, 0}, {60, 320, 0}, {60, 340, 0}, {60, 
  360, 0}, {60, 380, 0}, {60, 400, 0}, {60, 420, 0}, {60, 460, 
  0}, {60, 440, 0}, {60, 480, 0}, {60, 500, 0}, {65, 0, 74.2594}, {65,
   20, 0.26026}, {65, 40, 0.020016}, {65, 60, 0}, {65, 80, 0}, {65, 
  100, 0}, {65, 120, 0}, {65, 140, 0}, {65, 160, 0}, {65, 180, 
  0}, {65, 200, 0}, {65, 220, 0}, {65, 240, 0}, {65, 260, 0}, {65, 
  280, 0}, {65, 300, 0}, {65, 320, 0}, {65, 340, 0}, {65, 360, 
  0}, {65, 380, 0}, {65, 400, 0}, {65, 420, 0}, {65, 440, 0}, {65, 
  460, 0}, {65, 480, 0}, {65, 500, 0}, {70, 0, 79.4559}, {70, 20, 
  1.60128}, {70, 40, 0.0800641}, {70, 60, 0.020004}, {70, 80, 0}, {70,
   100, 0}, {70, 140, 0}, {70, 120, 0}, {70, 160, 0}, {70, 200, 
  0}, {70, 180, 0}, {70, 220, 0}, {70, 240, 0}, {70, 260, 0}, {70, 
  280, 0}, {70, 320, 0}, {70, 300, 0}, {70, 340, 0}, {70, 360, 
  0}, {70, 380, 0}, {70, 400, 0}, {70, 420, 0}, {70, 440, 0}, {70, 
  460, 0}, {70, 480, 0}, {70, 500, 0}, {75, 0, 86.9348}, {75, 20, 
  7.30146}, {75, 40, 1.96196}, {75, 60, 0.720144}, {75, 80, 
  0.46046}, {75, 100, 0.160096}, {75, 120, 0.080016}, {75, 140, 
  0.020008}, {75, 160, 0.020008}, {75, 180, 0.020012}, {75, 200, 
  0.020012}, {75, 220, 0.020012}, {75, 240, 0}, {75, 260, 0}, {75, 
  300, 0}, {75, 280, 0}, {75, 320, 0}, {75, 340, 0}, {75, 360, 
  0}, {75, 380, 0}, {75, 400, 0}, {75, 420, 0}, {75, 440, 0}, {75, 
  460, 0}, {75, 480, 0}, {75, 500, 0}, {80, 0, 91.4783}, {80, 20, 
  13.1653}, {80, 40, 7.70308}, {80, 80, 4.0208}, {80, 60, 
  5.48439}, {80, 120, 2.84114}, {80, 100, 3.22064}, {80, 140, 
  2.60104}, {80, 160, 1.80216}, {80, 180, 1.86037}, {80, 200, 
  1.36054}, {80, 220, 1.24149}, {80, 240, 1.08065}, {80, 260, 
  1.04021}, {80, 280, 0.960384}, {80, 300, 0.860861}, {80, 320, 
  1.0004}, {80, 340, 0.80016}, {80, 360, 0.720432}, {80, 380, 
  0.540432}, {80, 400, 0.480673}, {80, 420, 0.40032}, {80, 440, 
  0.20008}, {80, 480, 0.380076}, {80, 460, 0.460184}, {80, 500, 
  0.460276}, {85, 0, 94.2188}, {85, 20, 22.0332}, {85, 40, 
  12.2649}, {85, 60, 9.26371}, {85, 80, 7.78156}, {85, 100, 
  7.14572}, {85, 120, 5.84117}, {85, 140, 5.58447}, {85, 180, 
  4.90196}, {85, 160, 5.32106}, {85, 200, 4.5209}, {85, 220, 
  4.5209}, {85, 240, 4.74095}, {85, 260, 4.18167}, {85, 280, 
  3.80076}, {85, 300, 3.92392}, {85, 320, 3.72223}, {85, 340, 
  3.64073}, {85, 360, 3.38068}, {85, 380, 3.5007}, {85, 400, 
  3.36336}, {85, 420, 3.14126}, {85, 440, 3.02181}, {85, 460, 
  2.70054}, {85, 480, 3.02242}, {85, 500, 2.90174}, {90, 0, 
  95.7592}, {90, 20, 33.1333}, {90, 60, 18.1709}, {90, 40, 
  22.1689}, {90, 80, 15.8358}, {90, 100, 13.9856}, {90, 120, 
  12.7826}, {90, 140, 12.0272}, {90, 160, 10.024}, {90, 180, 
  10.8643}, {90, 200, 9.60192}, {90, 220, 8.80528}, {90, 240, 
  8.30332}, {90, 260, 8.64173}, {90, 280, 7.74465}, {90, 300, 
  7.40148}, {90, 320, 7.92317}, {90, 340, 7.48899}, {90, 360, 
  6.7427}, {90, 380, 6.82546}, {90, 400, 6.2425}, {90, 420, 
  6.76135}, {90, 440, 6.40512}, {90, 460, 6.84137}, {90, 480, 
  5.54222}, {90, 500, 6.16246}, {95, 0, 97.4795}, {95, 20, 
  42.9972}, {95, 40, 31.8191}, {95, 60, 28.6657}, {95, 80, 
  25.6903}, {95, 100, 22.8337}, {95, 120, 21.733}, {95, 140, 
  20.1201}, {95, 160, 18.6074}, {95, 180, 18.6875}, {95, 200, 
  17.3635}, {95, 220, 17.1869}, {95, 240, 16.6433}, {95, 260, 
  15.6062}, {95, 280, 15.3954}, {95, 300, 14.8118}, {95, 320, 
  15.3985}, {95, 340, 14.1657}, {95, 360, 14.9439}, {95, 380, 
  14.5229}, {95, 400, 14.0684}, {95, 420, 13.5681}, {95, 440, 
  13.4454}, {95, 460, 12.7051}, {95, 480, 13.5227}, {100, 0, 
  98.3397}, {95, 500, 13.2653}, {100, 20, 52.0416}, {100, 40, 
  43.1773}, {100, 60, 38.2153}, {100, 80, 34.9079}, {100, 100, 
  32.0328}, {100, 120, 30.5506}, {100, 140, 29.3729}, {100, 160, 
  29.0407}, {100, 180, 27.0908}, {100, 200, 26.4853}, {100, 220, 
  25.8252}, {100, 240, 25.245}, {100, 280, 24.1697}, {100, 260, 
  25.4152}, {100, 300, 25.2752}, {100, 320, 22.8646}, {100, 340, 
  23.0246}, {100, 360, 22.0888}, {100, 380, 22.2467}, {100, 400, 
  23.3247}, {100, 420, 22.9892}, {100, 440, 21.0242}, {100, 460, 
  20.9284}, {100, 480, 21.8775}, {100, 500, 19.6035}}

Which has dimensions

Dimensions[xyzValues]

{546, 3}

First, I have tried to plot these point in a contour plot:

ListContourPlot[xyzValues, PlotLegends -> Automatic, 
 ColorFunction -> "Rainbow", PlotRange -> {{0, 100}, {0, 150}}]

but this result does not reflect the data points (see for example there exist data in the blank zone of the plot). Probably, I am missing something, but the plot should be different. Plotting these points with another mathematical framework:

Observing these two plots, I do not know how to compute a good mathematical function to match these data point. I have made an approximation

model = a + b x + c x^2 + d Exp[-x] + e y + f y^2 + g x y + Exp[-y] h ;

fit = 
 FindFit[xyzValues, model, {a, b, c, d, e, f, g, h}, {x, y}]

{a -> -2.75705, b -> -0.196263, c -> 0.00570384, 
 d -> -4.9015, e -> 0.0131326, f -> 0.0000346097, g -> -0.000773403, 
 h -> 38.7327}

and plot this function:

Show[{Plot3D[Evaluate[model /. fit], {x, 0, 100}, {y, 0, 150}, 
   PlotStyle -> Opacity[0.8]], 
  Graphics3D[{Red, PointSize[.025], Map[Point, xyzValues]}]}]

However, it seems an approximation with much error. I would like to get a function with less error, if possible an expression that interpolates de data points.

Regards

Answer 1

Instead of FindFit you could use a linear interpolating function:

f = Interpolation[{{#, #2}, #3} & @@@ xyzValues, InterpolationOrder -> 1];

cols = RGBColor @@@ ({{53, 42, 134}, {15, 91, 221}, {18, 124, 215}, {6, 156, 207}, {21, 176, 180},
             {88, 189, 139}, {165, 190, 106}, {225, 185, 82}, {251, 205, 45}, {248, 250, 13}}/255);

plot = ContourPlot[f[x, y], {x, 0, 100}, {y, 0, 100}, Contours -> Range[10, 90, 10],
  ColorFunction -> (Blend[cols, #] &), PlotRange -> All, ContourLabels -> True, ImageSize -> 600]

The positions of the labels are not too good though. The code below replaces them by labels at manually picked vertices from each contour line.

plot /. Cases[plot, __Text, All] ->
  Catenate[MapThread[Thread[Text[#2[[1]], #[[1, 13 ;; ;; 76]]]] &,
    {Cases[plot, _Line, All], Reverse[Cases[plot, _Text, All]]}]]