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(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?

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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
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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

2 Answers 2

up vote 5 down vote accepted

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.

update: I've used this and it works great: https://pypi.python.org/pypi/powerlaw

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+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. –  dranxo 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. –  dranxo Apr 23 '12 at 8:47
1  
tuvalu.santafe.edu/~aaronc/courses/7000/csci7000-001_2011_L3.pdf –  dranxo Apr 25 '12 at 17:50

A few thoughts on your question:

  • How you report your model fit will depend very much on your audience, and your field. For example, in my field, model fit statistics like R^2 are very rarely reported - regarded as neither impressive nor particularly useful. Instead, some criteria for how you arrived at the model you arrived at tends to be described, and then you report your model results - we all assume you actually fit the model correctly.
  • "I happened across this form" is a bad explanation. A really bad one. Despite a fondness for stories of accidental genius, like the discovery of penicillin or quinine, "blind dumb luck" is not a reliable scientific process. For example, you have shown that that form is good at fitting your data, but you have not yet shown it is best at fitting your data. R^2 alone is not a sufficient metric for evaluating how well your model fits the data. See Anscombe's quartet.
  • As @rcompton mentioned, it looks like you're trying to fit a power law distribution without knowing it, but even if you do manage to well fit a power law, it's really best if you found some reason why you think its a power law. It may be enough to plot Y over time, head over to CrossValidated (or a college/department more comfortable with statistics) and systematically go over distributions that might give you roughly that look. There are others besides the power law distribution that might give you a superior fit.
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+1 for the insights. "systematically go over distributions that might give you roughly that look." - where can I find these? –  Erel Segal Halevi Apr 20 '12 at 2:57
    
@ErelSegalHalevi You might start at CrossValidated, this sites sister-site that concerns statistics and data analysis. –  Fomite Apr 20 '12 at 3:52

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