# Reporting curve-fit results in a scientific paper

(I hope this question fits this site; if not, accept my apologies).

I ran a certain simulation, and got a time series y(t), t = 0, 1, ... 20. After trying some functions, I found that:

y(t) =~ 1 / (A t + B)


Where A and B are coefficients I calculated using linear regression, with R^2 > 0.99.

What is the standard way to report such results in a scientific paper? Specifically:

A. I have no theoretic explanation, why the output looks like this (I know it should be decreasing, and that it's bounded from below, but not much more). It was just a successful guess. Should I describe all other unsuccessful guesses that I tried?

B. Whenever I run the simulation, I get slightly different values of A and B. Should I just report a random run, or should I run the simulation many times and average the results? If so, how many times is enough?

-
What do you want to convey? What does each individual simulation represent? –  Bill Barth Apr 19 '12 at 13:03
It's a simulation of land ownership. There are N citizens and N land plots. Initially, each land-plot is given to a random citizen. Then, each year, each land is sold with a certain probability p, and if it is indeed sold, the buyer is selected at random. After 50 years, I run a "Jubilee" procedure where some lands are returned to the original owners, if these owners currently have no land. I measure the number of citizens without land (y) after each Jubilee (t). Certainly y(t) is non-increasing. I want to show that it is decreasing in a predictable rate, and that it converges to 0. –  Erel Segal Halevi Apr 19 '12 at 13:43
It seems to me that you should develop a statistical representation of $A$ and $B$, then (mean, median, etc.). –  Bill Barth Apr 19 '12 at 14:25
Consider a model with N+1 different species $x_n$ where $n=0\dots N$, which denotes concentration of landowners with $n$ plots of land. Now you can apply chemical kinetics theory to your problem. –  Deathbreath Apr 19 '12 at 17:48
Bill: do you mean that I should calculate A and B many times, then report the mean and std? I think a better approach is to do a single linear regression with all samples from all simulations. But how many times should I run the simulation? –  Erel Segal Halevi Apr 19 '12 at 18:06
show 1 more comment

You're trying to fit a power law to your distribution. Very interesting. These show up all the time in graph theory, social networks, and a slew of other places.

There's some tutorials on fitting your data here and here.

Also, in reference to question A., how does the probability of a person buying land depend on how much land they already have? You may be able to use Barbasi's model to explain why a power law is a reasonable fit to your data.

-
+1 for all the links! I also thought of power law, but it's simple form (y = A t^k) does not entail the form I found, because of the B constant (y = (A t + B)^-1). Is there a more general form? –  Erel Segal Halevi Apr 20 '12 at 4:34
If you are interested in describing the shape of the curve then you should factor and shift before fitting a power law. The fact that you have a B is not relevant to the shape of the curve. –  rcompton Apr 20 '12 at 9:07
Sorry, I didn't understand you, what do yo mean by "then you should factor and shift"? –  Erel Segal Halevi Apr 21 '12 at 17:57
Set x = t+B/A. Then (At+B)^{-1} = (A*x)^{-1} which is the form in the links. –  rcompton Apr 23 '12 at 8:47
tuvalu.santafe.edu/~aaronc/courses/7000/csci7000-001_2011_L3.pdf –  rcompton Apr 25 '12 at 17:50