I should have some references to other analyses of the Iwo Jima data, and probably the day-to-day troops levels too. Give me a few days (busy weekend!) and I’ll try to post something relevant.

I’ve been getting some “500″ server errors when I try to load your blog. I’m not sure what that means, but I thought I should let you know.

—Dan

I have some simple simulations set up in a spreadsheet, intended for a post on my blog, but I want to set up something a little better in R. The spreadsheet needs some more work before it’s ready to share, but I’ll show it to you when it’s done.

Again, many thanks for this discussion. :-)
—Dan

My approach: simpler! I don’t consider the probability that is an outcome. Rather, I consider the probability of being alive (in army X or army Y) at the next time-step. This is easy because at the next time step we can calculate exactly how many people have died, from the actual differential equations (or approximate such). However there’s a problem, in that the matrix changes with time, which is an approach we don’t usually take in the Markov treatment (the transition matrix is usually fixed in time, so that powers of such a matrix, another matrix, tells us the distribution of a start vector after a time step equal to the power of the matrix). In my approach, we essentially have to drag the initial vector across a changing matrix, to obtain the trajectory of a start vector. So perhaps it is not a Markovian matrix after all (since we can only take powers of such if the time component drops out), but a discretization of the differential equations in a convenient (matrix) form, which I reasoned out thinking Markovianly.

As to the expected value and variance of the Markov model in the pdf you provided, I’m not sure what you mean by it being a geometric distribution (but I did catch the reference to a geometric distribution when it talks about fractal models). At worst, you can probably simulate the Markov model and calculate the expectation and variance, like this: pick a suitable bijection of the (allowable) coordinates to the positive integers (your new, y’-axis, there may be a question about how to order them, perhaps, although my hunch is use a diagonal method?), and remember that there will be several absorbing states, namely , where m reds and n greens are the start coordinates. Pick a time step (1-unit), and let unit increases be your new x’-axis (we want to visualize the actual Markov process). Next go to your start coordinate. Run the model and see how it ends (in other words, get a realization of the process, it ends when it hits an absorbing state). Calculate the expectation and variance (?) of the realization. Repeat many times and review point-estimators to figure out how to make sense of this data.

I don’t recall actually seeing a treatment of expectation and variance of a Markov process, although I agree I should know it/investigate it further; the preoccupation is usually more about steady-state probabilities (in regular matrices, the higher-power matrices stabilize, which means that an initial vector will end up as a steady-vector at infinite steps of time). Thus, since any vector will eventually result in the steady-state (for regular matrices anyway), the expectation of the process (ensemble), as we take more and more (infinite) unit time steps, should be the inifinite-step probability steady-state vector dot the state vector (in other words, weigh the states by their respective steady probabilities). This however, is different from what I think you mean; from this viewpoint, you are getting an expected state, the intermediate state of all paths, which may not be an end-state (an outcome of the process, or, who wins on average). The variance of the process can probably be as great or as small depending on your state space and on whether states communicate with each other, so it may be realization-dependent. (Hence why we don’t usually talk about variance in this context).

Does this help??

I think the pdf is really interesting, I’ll write a post with a simulation, it may make things clearer.

If I were to ask for help, it would be in calculating the variance of a Markov model. For instance, what is the variability of the results in the SIR model (on the model above). A nagging voice in my head tells me this ought to be analogous to calculating the variance of the geometric distribution, and therefore something I could do if I understood more about Markov chains. On the other hand, the variance at a given time will be dependent on the previous time, leading to a distribution of all possible outcomes at time t, and that seems like it might be a much harder problem.

Now that I’ve written all this, I’m beginning to see I may be misunderstanding what you have really done in the Markovization above, which may render my question meaningless. I should study! :-)

If this is of any interest to you, here is a link to a fairly recent paper on Lanchester attrition models, which also gives a Markov version of the model:
http://dspace.dsto.defence.gov.au/dspace/bitstream/1947/4233/1/DSTO-TR-1822%20PR.pdf
I also find this paper much more readable than most of the other literature on the topic.

Gladly! :-)
Recently I’ve been writing about Lanchester’s Laws, and I have been wanting to show the Markov or probability based version to demonstrate the inherent uncertainty of the results of this sort of prediction. I could do this demonstration in a spreadsheet, but I knew there had to be a more rigorous way to show it. I’ll use this to better understand the games I like play, and maybe (eventually) to help design a game of my own.

More generally, I’m a biostatistician and wargamer with a hobby interest in the mathematics of games, which is the general topic of my blog. I’m pretty good at applied math and stats, but my education is lacking on some topics and I’ve never been great at theory. I can read the journals though (and math blogs!) , so I search for others writing about related topics to fill the gaps in my own knowledge.

I now have you on my RSS feed, added to my blog roll, and I’ll be linking to this in an upcoming blog post. No doubt you will be see more of me in the future. :-)
—Dan