Multivariable non-linear fitting

Question

I have a function called "C" that I know it depends on 6 variable. I have a set of data points, 6 rows by 9 columns, in which in each row only one variable is changing and the rest of the variables are constant.

I was able to fit each row independently (one variable fitting and modeling) and I double checked on excel as shown below:

I would like to ask about the best way of fitting the function "C" as a function of all the 6 variables into a single equation.

For the single variable fitting, this is the code I used:

    data1 = {{2, 4.8128*10^-12}, {3, 7.5623*10^-12}, {4, 
   9.9060*10^-12}, {5, 1.1378*10^-11}, {6, 1.3474*10^-11}, {7, 
   1.5636*10^-11}, {8, 1.7803*10^-11}, {9, 2.0028*10^-11}, {10, 
   2.2438*10^-11}}; (* x1 and C data pairs*)
g1 = ListPlot[data1] 
g2 = LinearModelFit[data1, x, x] 

g22 = Normal[g2] 
Show[Plot[g22, {x, 2, 10}], g1] 
g2["ParameterConfidenceIntervalTable" ]

Answer 1

I have copied the data from your notebook and would like to share some information.

Fit Individual Components

In order to facilitate the fitting and plotting I created groups of six {x,y} pairs, one group for each of the six variables.

dataGroup = 
  Map[Transpose@
     Join[{data7[[1 + (# - 1)*9 ;; #*9, #]], 
       data7[[1 + (# - 1)*9 ;; #*9, 7]]}] &, Range[6]];

varNames = {"Fingers", "Width", "Spacing", "Length", "Gap", 
   "Thickness"};

Next fit the individual groups using the models from your spreadsheet.

model = ConstantArray[0, 6]; param = ConstantArray[0, 6]; sol = 
 ConstantArray[0, 6];

model[[1]] = a1 x + b1; param[[1]] = {a1, b1};
model[[2]] = a2 Log[x] + b2; param[[2]] = {a2, b2};
model[[3]] = a3 x^-b3; param[[3]] = {a3, b3};
model[[4]] = a4 x + b4; param[[4]] = {a4, b4};
model[[5]] =  a5 x + b5; param[[5]] = {a5, b5};
model[[6]] = a6 x^2 + b6 x + c6; param[[6]] = {a6, b6, c6};

Use FindFit along with the arrays just created.

(sol[[#]] = 
    FindFit[dataGroup[[#]], model[[#]], param[[#]], x]) & /@ Range[6]
(* {{a1 -> 2.1325*10^-12, b1 -> 8.7591*10^-13},
    {a2 -> 2.65771*10^-12, b2 -> 3.22181*10^-11},
    {a3 -> 7.84083*10^-13, b3 -> 0.413135},
    {a4 -> 3.69143*10^-10, b4 -> -2.19223*10^-13},
    {a5 -> -6.16*10^-11, b5 -> 1.4081*10^-11},
    {a6 -> 4.59524*10^-6, b6 -> 2.38786*10^-9, c6 -> 1.20616*10^-11}} *)

Plot the results

Grid@Partition[Module[
     {
      fun = model[[#]] /. sol[[#]],
      xmin = Min[dataGroup[[#, All, 1]]],
      xmax = Max[dataGroup[[#, All, 1]]]
      },
     Show[
      ListPlot[
       dataGroup[[#]],
       PlotStyle -> {PointSize[Large], Red},
       PlotLabel -> varNames[[#]]
       ],
      Plot[fun, {x, xmin, xmax}, PlotStyle -> Black],
      PlotRange -> All,
      ImageSize -> 300
      ]
     ] & /@ Range[6],
  2]

Data problems

When an attempt was made to fit the data in one fell swoop, problems arose. This is due to three problems in the data.

Presumably all the variables were held constant and one allowed to vary in each of the six groups.

The constant inputs I will call the center point and equal:

data7[[5]]
(* {6, 1/1000, 1/1000, 19/500, 1/500, 1/5000, 1.3474*10^-11} *)

The first problem was relatively minor. For variable three (Spacing) the value of variable two (Width) is 0.0015 when it should have been 0.001.

More serious problems are present for variable five (Gap) and variable six (Thickness). They have the same input variables but a radically different capacitance.

{data7[[5]], data7[[37]], data7[[48]]}
(* {{6, 1/1000, 1/1000, 19/500, 1/500, 1/5000, 1.3474*10^-11},
    {6, 1/1000, 1/1000, 19/500, 1/500, 1/5000, 1.3948*10^-11},
    {6, 1/1000, 1/1000, 19/500, 1/500, 1/5000, 1.2764*10^-11}} *)

New synthetic data was created from the existing data and repaired using these steps:

group 3 (Spacing) - make variable 2 = 1/1000 and subtract from y so that it matches at the center

group 5 (Gap) - subtract from the y values so that there is a match at the center point

group 6 (Thickness) - add to the y values so that there is a match at the center point

dataF = data7;
dataF[[19 ;; 27, 2]] = 1/1000;
dataF[[19 ;; 27, 7]] = data7[[19;;27,7]] + (data7[[5,7]] - data7[[19,7]]);
dataF[[37 ;; 45, 7]] = data7[[37;; 45,7]] + (data7[[5,7]] - data7[[37,7]]);
dataF[[46 ;; 54, 7]] = data7[[46;;54,7]] + (data7[[5,7]] - data7[[48,7]]);

dataGroupF = 
  Map[Transpose@
     Join[{dataF[[1 + (# - 1)*9 ;; #*9, #]], 
       dataF[[1 + (# - 1)*9 ;; #*9, 7]]}] &, Range[6]];

Fit all of the data using center point

Fit the data in one fell swoop by using the form for the individual components and deviations from the center point. FindFit and NonlinearModelFit have the same input form. The latter produces a FittedModel and provides statistical information on the parameters.

Note: I am not recommending that this be the final solution. It would be much better if a valid theoretical model could be found.

My purpose here is to show the mechanics of how to optimize all of the data.

xF = data7[[5, 1 ;; 6]]; yF = data7[[5, 7]];

modelF = a1 (x1 - xF[[1]]) + a2 (Log[x2] - Log[xF[[2]]]) + 
   a3 (1/x3^b3 - 1/xF[[3]]^b3) + a4 (x4 - xF[[4]]) + 
   a5 (x5 - xF[[5]]) + a6 (x6 - xF[[6]])^2 + b6 (x6 - xF[[6]]) + yF;

Use NonlinearModelFit

solF = NonlinearModelFit[dataF, modelF, 
   paramF, {x1, x2, x3, x4, x5, x6}];

(* {{a1 -> 2.1325*10^-12}, {a2 -> 2.94108*10^-12},
    {a3 -> 1.12959*10^-13, b3 -> 0.657385},
    {a4 -> 3.7597*10^-10}, {a5 -> -6.07353*10^-11},
    {a6 -> 3.88793*10^-6, b6 -> 4.23126*10^-9}} *)

Now plot the results

Grid@Partition[Module[
     {
      args = xF,
      xmin = Min[dataGroupF[[#, All, 1]]],
      xmax = Max[dataGroupF[[#, All, 1]]]
      },

     args[[#]] = x;

     Show[
      ListPlot[
       dataGroupF[[#]],
       PlotStyle -> {PointSize[Large], Red},
       PlotLabel -> varNames[[#]]
       ],
      ListPlot[{{xF[[#]], yF}}, PlotStyle -> {PointSize[Large], Blue}],
      Plot[solF[Sequence @@ args], {x, xmin, xmax}, 
       PlotStyle -> Black],
      PlotRange -> All,
      ImageSize -> 300
      ]
     ] & /@ Range[6],
  2]

Information on the parameters

solF["ParameterTable"]