Question for stats nerds--stringing together poisson processes?

[scrub]

Hey there guys.

I'm trying to write a stochastic model of a biochemical system for my thesis so that I can graduate and my parents don't kill me.

I'm describing the concentrations of the various species in the system as a set of coupled stochastic equations, and solving them using a monte carlo algorithm.

Problem is that I really don't have the computing power or time to write code for each individual intermediate in polymerization reactions.

I'm not sure if it's valid to do this, but I've been thinking about having one reaction for the beginning of the polymerization, then querying some sort of time distribution for the polymerization and having the model wait to update the state of the those molecules until after the time the distribution returns for the polymerization.

Anyway, I'm not really much of a statistician, but a friend reccomended approximating the time it takes for a single step in the process as a poisson distribution, and then determining the distribution of the time it takes for N consecutive poisson processes to complete.

Is there a general form for this distribution that is widely known and/or could someone reccomend a good textbook/resource to try to figure this out?

Thanks!

scrub

[beerbandit]

I don't think that calling someone a nerd is a good thing to do before asking for help.

[scrub]

[ QUOTE ]
I don't think that calling someone a nerd is a good thing to do before asking for help.

[/ QUOTE ]

Thanks for the advice.

If you actually had anything of value to say on the subject and decided not to reply because I facetiously directed my post about an inherently nerdy topic to "stats nerds", then you need more help than this nerd can possibly help to provide by apologizing.

But, please accept my apology nonetheless.

On the other hand, if you were just making a vapid reply to bump up your post number...

scrub

[pzhon]

I'm describing the concentrations of the various species in the system as a set of coupled stochastic equations, and solving them using a monte carlo algorithm.

Funny, I've been doing something that could be described similarly to estimate equities at the final table.


Anyway, I'm not really much of a statistician, but a friend reccomended approximating the time it takes for a single step in the process as a poisson distribution, and then determining the distribution of the time it takes for N consecutive poisson processes to complete.

This sounds confused. A Poisson distribution is a count, and has a positive probability of being 0. It doesn't scale well as you change your time steps. Do you want an exponential distribution instead?

I think you want the gamma distribution which generalizes the exponential distribution. It is the sum of N independent exponential distributions, or the infimum of the times that a Poisson process has a value of at least N.

If you are using time steps of a fixed size, there are discrete analogues. I think the analogue of the exponential distribution is the geometric distribution. I'm not sure what the analogue of the gamma distribution is; IIRC the hypergeometric distribution is something else.

A problem with using the gamma distribution is that the steps within your reaction are not identical. If this is important, then you can convolve different exponential distributions, or perhaps a few gamma distributions. The integrals look easy.

If you actually want the sum of independent Poisson distributions, that's also a Poisson distribution.


Is there a general form for this distribution that is widely known and/or could someone reccomend a good textbook/resource to try to figure this out?

A good resource for math questions like this is google.com, since many course notes have been put on the web.

[scrub]

Thanks for the reply!

Like I said--I don't know very much about this stuff. I guess I really wanted an exponential distribution. I'm pretty sure that the approximation I wanted to use would have been frowned by the powers that be anyway--I certainly don't know enough about the mathematics of this stuff to try to justify doing it that way.

While the reactions are not the same, I don't get the impression that the individual steps are experimentally characterized anywhere near well enough to write differeent rate constants for them. I found a paper that modeled growing polymers with three "states"--starting, growing, and ready to finish--using counters to track the increments between the states, and separate rate constants for each reaction. Doing it this way will let me have an "exact solution," and adding three equations to my model is not horrendous in terms of processing time.

I've been thinking about how you would write a decent model for a final table. Are you modeling hand EVs or player win rates, or are you actually trying to model the play of individual hands?

Anyway, thanks for taking the time to respond to such a muddled question.

scrub

[pzhon]

[ QUOTE ]
I've been thinking about how you would write a decent model for a final table. Are you modeling hand EVs or player win rates, or are you actually trying to model the play of individual hands?

[/ QUOTE ]

I would like to estimate the probability that each stack finishes in each place. If you assume that this does not depend on the exact payout structure, then you can determine the proper deals at the final table. For example, if 1st plays 30%, 2nd pays 15%, 3rd pays 10%, and 4th pays 5% (total 60% in the last 4), what is the fair way to split the prizes among the last 4 players, with 80k, 40k, 10k, and 10k chips? Once you know the theoretical values, you can quantify how much nominal EV (expected chips) you should sacrifice to increase your share of the prizes.

If you accept the simplifying assumptions that position does not matter and that the amount at stake in each hand is alway small compared with the stacks, it is still a tough problem. In addition, I would like to incorporate the possibility of occasional larger bets, such as hands moving in. I doubt this constitutes a decent model of the final table, but it would be a start.

A naive formula for the probability with which you place second is the sum over all of your opponents X of Probability(X wins)*Probability(you win if X is kicked out). This formula is not very accurate, according to simulations, but it is better than nothing. Does anyone have better approximations?

[Bozeman]

check out

4 way (http://forumserver.twoplustwo.com/showflat.php?Cat=&Board=tourn&Number=519924&Forum= All_Forums&Words=fourway&Match=Entire%20Phrase&Sea rchpage=0&Limit=50&Old=6months&Main=519924&Search= true#Post519924)

and the thread referenced therein.

[pzhon]

Wonderful! Thanks for your work and references. I had gotten to the conformal transformation of the triangle to a disk for the case of three players, but IIRC there is no analogue in higher dimensions since conformal transformations are rare, so I used numerical methods, too.

I don't have a suggestion for a good family of functions to fit to the probabilities/equities. Such a best-fit function may be a good way summarize numerical results.

I'm surprised that the independent chip model agreed as well as it did with your simulations. I believe it fails more radically for larger numbers of players and more lopsided distributions. It would be nice to contrast these models and the formulas people use in practice.

In addition to recommending deals, these models can be used to analyze whether to sacrifice expected chips to try to knock a player out. Eliminating an opponent is remarkably overvalued by many players. However, if people play aggressively at the final table, then the diffusion model may underestimate the benefit from eliminating an opponent.

By the way, the gambler's ruin in higher dimensions arises in the analysis of backgammon, too. There are 6 possible outcomes of a cubeless game. The use of the doubling cube tends to decrease the value of the bonus points (gammons and backgammons), but the extent to which this happens is not clear. There is also no good model for the value of a position that is too good to double.

[daryn]

scrub, get your hands on a statistical physics book, that may help. i worked out a similar problem using monte carlo simulations to simulate the ising model of thermodynamics.

[BillC]

I think your question is more in probability theory more than stats. You have to be aware of a stachastic processes, as opposed to sets of distributions. You might try a binomial tree model as a discrete approximation. When you say 'stochastic equations', do you mean 'stochastic differential equations'? What are they?
You really need to learn some prob. theory to mess with this stuff.
Your choice of distribution depends on the nature of the problem (Poisson vs binomial (discrete); gamma va exponential (continuous)).

BillC,
Math nerd, Ph.D.