Fitting multiple datasets to the same nonlinear model

Question

Perhaps a rather simple question, but the suggested questions seem to all be covering something slightly different. What I have can be described as follows.

One takes a dataset, call it Data; it has dimensions 10 by 1000 by 2 lets say, which is to say it is 10 arrays of {x,y} data of length 1000.

Now I know the following about it: it behaves according to a model Model[x,a,b,c]. Moreover, all 10 arrays have the same a,bbut not the same cparameter. What I am interested in is how one does a NonLinearModelFit over such a dataset, fitting a,b,c1...c10 in a single go. I should note that I have an initial value array cinit of dimensions 10 as well.

One way I thought of that should in principle work is with KroneckerDelta; I could prepend each dataset with some coordinate that I could use in addition to x and then make a model with the deltafunctions, but this feels very sloppy. There should be an easy way, should there not?

I would post the full model and the data I am working with, but not only is it a big dataset, the parameters and the model are still very much undetermined. I think it would make the question rather offtopic as it is my job to figure out the parameters, not you. So I thought it would make sense to just stick to the concept.

Answer 1

Here is a crude way to do it. First the dataset id is appended to the 1000 x 10 x 2 data array. Then the function you have is modified to create a dummy variable vector and the dot product is used to select the parameter of interest: c1, c2, c3,...., or c10.

(* Number of different values of c *)
nc = 10;
(* Number of observations per dataset *)
n = 1000; 

(* Create some data for a 1000 x 10 x 2 array *)
xx = Table[i/n, {i, n}];
data = Table[
   Transpose[{xx, 
     5 + 2 xx + xx^2 + RandomVariate[NormalDistribution[0, 1], n]}], {i, nc}];

(* Add in dataset number: 1 through nc *)
data2 = Flatten[
   Table[{i, data[[i, j, 1]], data[[i, j, 2]]}, {i, nc}, {j, n}], 1];

model[x_, a_, b_, i_, c_] := Module[{d, j},
   (* Create dummy variables *)
   d = Table[If[j == i, 1, 0], {j, Length[c]}];
   (* Determine function value *)
   a + b x + (c.d) x^2];

nlm = NonlinearModelFit[data2, 
   model[x, a, b, z, nc, {c1, c2, c3, c4, c5, c6, c7, c8, c9, c10}],
   {a, b, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10}, {z, x}];
nlm["BestFitParameters"]
(* {a -> 4.99484, b -> 1.93603, c1 -> 1.18239, c2 -> 1.0829, 
 c3 -> 1.21282, c4 -> 1.06046, c5 -> 1.13936, c6 -> 1.07549, 
 c7 -> 1.01086, c8 -> 0.960816, c9 -> 0.979848, c10 -> 1.03909} *)

But note that I get a warning:

Experimental`NumericalFunction::dimsl: {x} given in {z,x} should be a list of dimensions for a particular argument.

Also, besides assuming common values for a and b , you are also assuming a common error variance. Residuals should be checked to see if there are any departures from that assumption or if a more complex error structure is warranted.