Fitting parameters of user defined function returning discrete values

Question

So, I'm fairly new to Mathematica (apologies in advance for the noobiness), and I'm having some problems regarding fitting the parameters of a user defined function to data. I'll give a representative toy example of what I'm trying to accomplish.

Imagine that I have data that is comprised of two lists:

TestData = {
  {1, 0, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11, -12, -13, -14, -15},
  {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}
};

And I have a model/function that itself produces discrete data such as this. This function would be something like:

TestFunction[A_, B_] :=  
   Module[{results = {}, vec1 = {}, vec2 = {}},   
      For[i = 1, i < 18, i++, Do;  
         [vec1 = Append[vec1, (A - i)];  
          vec2 = Append[vec2, (B + i)]]];  
      results = Append[results, vec1];  
      results = Append[results, vec2];  
      results]

Now, I want to find the parameters A and B that give the best approximation to TestData (i.e., such as the distance between TestData and the results of the function would be minimized).

I tried looking into the FindFit and NonlinearModelFit functions, but these would seem to be more suited for continuous functions(?), so I'm not sure how to use it with a function that returns specific and discrete values. How would you go about solving this question?

Thank you very much in advance.

Answer 1

First of all, two general remarks.

1. Try to avoid using procedural loops (For, While etc.) in Mathematica. Instead use functions like Map (/@), Array, Apply (@@) or Table.
2. You should always use lower case letters for your own functions and variables.

The Do in your TestFunction is either missing a second argument, or (judging from the structure of your test data) shouldn't be there at all. I assume that you want the function to work like this:

testFunction[a_, b_] := Transpose@Table[{a - i, b + i}, {i, 17}]

Since the first list in testFunction's result depends only on a and the second only on b, respectively, you can find the values for a and b seperately. Nevertheless, here's a piece of code that minimizes the norm of the difference for a and b in one step:

Minimize[Plus @@ Norm /@ (testFunction[a, b] - testData), {a, b}]

{0, {a -> 2, b -> 3}}

The result is the minimum of the norm and a list of replacement rules for a and b.

For more complicated data or functions, it probably makes sense to use NMinimize instead.