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  #31  
Old 07-17-2009, 12:45 AM
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Originally Posted by Actuarialsuck View Post
I may be completely off base here but couldn't you use strange attractors to generate random numbers? I think I recall someone talking about that in college but I could be just making it up... atomic to the rescue?
Certain chaotic dynamical systems have the appearance of randomness, but chaos is not necessarily a good criterion for choosing pseudorandom numbers. The reason is that these systems can exhibit global structure within the apparent randomness. Chaos in dynamical systems is characterized by "sensitive dependence on initial conditions" (a.k.a. the "butterfly effect"). But that is not a sufficient condition for statistical randomness.

For example, I read E. N. Lorenz's original paper "Deterministic Nonperiodic Flow" back in college and one of the diagrams that struck me was one in which he plotted something like the successive local maxima of the Lorenz equations. This was a long time ago so I can't remember the exact thing he plotted, but it formed a very nice smooth inverted "v" curve with a sharp cusp in the center. That is anything but random behavior--it is order hidden within chaos.

It stands to reason therefore, that the choice of chaotic dynamical system for a PRNG would need to undergo rigorous statistical analysis before it could be used as such, because it is not readily obvious whether such a system could yield some underlying order if you looked at its behavior in some very specific fashion.

Finally, I also want to point out that the implementation of strange attractors in programming code is not necessarily more efficient compared to existing PRNGs based on, say, cellular automata (this is the method used by Mathematica).
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  #32  
Old 07-17-2009, 12:47 AM
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Hmm that's very interesting, I was referring to somehow linking in the randomness as to how many loops you make around each node of the attractors if that makes any sense.
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  #33  
Old 07-17-2009, 07:24 AM
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Is this as dumb as I think it is? I keep reading it trying to decipher how that solves anything.
Yes it is. It has been mentioned in the past that MS often makes it clear that they don't use Excel for any serious work. This is just one more clue along that line.
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  #34  
Old 07-17-2009, 07:27 AM
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One also has to be very careful about running out of randomness. I had a colleague who spent some number of weeks trying to debug a monte carlo simulation, only to eventually find out the randon number generator wasn't "big" enough for the number of simulations. It was starting to create patterns, which was screwing up the distributions of the results, or something like that. I don't know how good the excel rand is in this respect- probably if you are doing something in excel, you don't have so many experiments to worry about it.
Your last sentence is exactly what this thread is discussing - for doing Monte Carlo analysis, Excel's Rand() function is far from adequate.
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  #35  
Old 07-17-2009, 07:35 AM
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I've been thinking of writing something up on floating point computation issues for CompAct, or maybe even one of the more technical actuarial journals.
Please do it. This was one of the things that I believe every actuary should know!

TTIA Glad you started it.
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  #36  
Old 07-17-2009, 10:19 AM
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No, it's not Mersenne Twister. [Not the one incorporated in the C3 Phase III interest rate scenario generator. I'm currently working on it with ESIWG.....which is the sequel to ESWG.]
Thanks for the clarification. I see that in the comments it says it is not MT.
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  #37  
Old 07-17-2009, 10:24 AM
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This is the kind of thread that really belongs in an actuarial wiki.
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  #38  
Old 07-17-2009, 10:45 AM
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This is the kind of thread that really belongs in an actuarial wiki.
Later, later.

Just a few comments on other random digit sources:
Of course the strange attractors from nonlinear dynamical systems [those "chaotic" sequences] are just as deterministic as a regular linear congruential PRNG. [Using radioactive decay of particles is a nifty random digit generation technique, though one still needs to check the stats on the sequence.... gotta make sure God ain't throwing loaded dice. ]

The reason that the linear congruential PRNGs are popular is that they're easy to code, and one knows the repeat cycle and distribution of numbers over that repeat [which covers all possible integers in the cycle] due to number theory. For any given nonlinear dynamic system, it's unlikely you're going to get a uniform distribution, but that's okay as long as you know what the distribution is... then you can "undo" it to force it into a uniform or whatever other distribution you desire.
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  #39  
Old 07-17-2009, 11:37 AM
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Later, later.
And of course that was meant as a compliment.
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  #40  
Old 07-17-2009, 11:42 AM
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The reason that the linear congruential PRNGs are popular is that they're easy to code, and one knows the repeat cycle and distribution of numbers over that repeat [which covers all possible integers in the cycle] due to number theory. For any given nonlinear dynamic system, it's unlikely you're going to get a uniform distribution, but that's okay as long as you know what the distribution is... then you can "undo" it to force it into a uniform or whatever other distribution you desire.
Whups, and I meant to say that for some of these strange attractors, a closed form for the resulting distribution doesn't exist [not always true, just sometimes], so then you'd have to use some sort of histogram results to "undo" the transform.... and that causes practical problems and any benefit from using said strange attractors is now gone. To use the strange attractors over the standard PRNGs, you need to show that they're better in some way - such as you get the same sort of stats on the results, but require fewer computations, or there are certain tests they satisfy that aren't satisfied by the regular PRNGs.
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