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I have a special type of model fitting problem that I am trying to solve in Mathematica. Any insight how to attack it using available tools of Mathematica will be great to hear.

We have a variable $$ H=\{h_k\}, \text{ where } k=1,...,15, $$ and each $h_k$ is a 4-tuple of complex numbers $$ h_k=\{a_j+i b_j\} \text{ where } j=1,...,4 $$ and $a_j,b_j$ are normally distributed real numbers.

Now this $H$ is mapped to a object $P=\{p,c\}$ where $p$ is a permutation of integers $\{2,3,...,16\}$ and $c$ is a real number. Now we seek to find an analytic model $f$ such that $f:H \rightarrow P$. Through a Monte Carlo simulation I have got 50000 data for this map $f$. I am stuck with the question about what kind of analytical model to choose and which fitting procedure will be best if one wants to solve this problem in Mathematica? Hope to hear from you people.

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Welcome, PlatoManiac. LaTeX formatting is enabled on this site. Also, there are some conventions in place from the private beta which you may want to review. Good luck. –  rcollyer Jan 26 '12 at 15:06
    
I think will get better answers if you explain a little more about the problem. Are you seeking an analytical form that you can fit to this? Hoe do you arrive from a matrix of complex numbers to a permutation? (Is it the ordering of another set of 15 real numbers?) What does $c$ mean? –  Szabolcs Jan 26 '12 at 15:15
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Think about it like this: If you asked "I have a function $f:\mathbb{R} \rightarrow \mathbb{R}$, what kind of analytical model can I fit it to?", it would still seem very vague and much too general. Can you be more specific? What answer would you give to my example question? –  Szabolcs Jan 26 '12 at 15:19
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@PlatoManiac Ideally, 1. you come up with an idea for $f$ (call it $f^*$) which has parameters $\alpha_1, \alpha_2, ...$, and implement a function that calculates it 2. You define a distance function on $P$ 3. you minimize the distance function wrt $\alpha_1, \alpha_2, ...$ so $f^*$ will give a result as close as possible to the data. But you probably know this. I think this is not really a Mathematica question, but a method-question. –  Szabolcs Jan 26 '12 at 17:10
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DataModeler from evolved-analytics.com is intended for this sort of problem. It is a commercial product built on Mathematica. They have a 60-day trial version and also an academic license option. (I realize this constitutes an advertisement, but the fact is that the product is intended for specifically this type of problem.) –  Daniel Lichtblau Jan 26 '12 at 18:24

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