Null Disquisition

I can’t believe I’m actually doing work this afternoon instead of playing Spore. Oh well, I’m already up to Civilization Phase (only took 12 hours). The wife is at Starbucks off campus studying with her cohorts (with shitty T-Mobile wifi), so I went to the Starbucks on campus (with awesome free campus Wifi).

Earlier this week, I promised that I was going to keep everyone updated with how my research is going and what I’m doing. So, here we are.

As previously mentioned, this first paper I’m writing is a “Why and When” sort of paper. We talk about three popular methods for doing MCMC simulations and which is best under what circumstances. The three methods are traditional Molecular Dynamics, Multiensemble, and Hamiltonian Replica Exchange Molecular Dynamics. The method I’m researching is the last one, hREMD (love the title). You can tell it’s a recent method cause the name is really long and convoluted. I’m pretty sure all the good names in MCMC were taken by the mid 90s. I won’t go into full detail on each method here (trying not to lose anyone’s interest), just know they exist.

The basic jist of the paper follows. You can break down the computation resources of a simulation into two parts: equilibration phase, and production phase. When doing MCMC, you must let your system fully equilibrate before you can start sampling data. An example of why this is would be, in 2D suppose you stick a particle in a box and let the particle move around a tiny bit each iteration. For the next several iterations, the position of the particle is going to be correlated to the starting point. This is called configuration bias (or startup bias, et al.). The following figure is the Autocorrelation of some MC time series data (the thickness is from the error bars).

A visual analogy follows: Suppose you take a thatch of color, magenta. If you break it down into 3 color channels (red, green, blue) the corresponding hex code would be something around #A03. Call this our starting point.

Iteration 1 (#AA0033)

Now let’s make up an update rule for our Markov chain. Each iteration, we pick a color channel and shift it by some amount x where x is a random integer between [-1,1]. After 100 iterations, we have moved around in the 3d color-space (where each channel is a dimension), and have ended up at #953.

Iteration 100 (#995533)

Hmm. Not much has changed. Lets look at 10000 iterations.

Iteration 10000 (#222255)

Ok, that’s better. What I’m trying to demonstrate is that when you do a stochastic simulation like this, the starting point is going to bias what the system does for the first several iterations. You need to let the system run for a very long time in order for your current state to have no “history” of the first state. The plot above shows the correlation of the system as it goes through time. Notice, at the beginning, the correlation is very high (in fact at time 0 it is infinite). The reasoning for this is the same as for why Iteration 1 and Iteration 100 of our color simulation are very similar.

That said, my paper talks about how long it takes each of the 3 different methods to reach equilibrium - when the current state has lost all memory of the original state. I think I’ll make a little demo of the color thing.

-David