# Autocorrelation time

1. Dec 9, 2003

### Moetasim

Well, can anyone tell me about the significance of autocorrealtion time in statistcal analysis of some time series data especially in case of monte carlo simulation output??

2. Dec 9, 2003

### Ambitwistor

There is little in terms of hard proof that Markov Chain Monte Carlo simulations will produce statistically independent samples. The autocorrelation time is one heuristic tool people use to estimate whether they do.

The problem is that in an MCMC random walk, if you sample after every move, the samples will usually be highly correlated: a move produces a point in parameter space that is very close to the original point. Consequently, you end up underestimating the true error; the error doesn't decrease with the square root of the number of samples if the samples aren't independent of each other.

(If you take 100 measurements of some quantity, you'll improve your precision by a factor of 10 compared to if you take 1 measurements. But if you just write down 100 identical copies of the same measurement, you won't really have improved your precision by 10 --- you won't have improved it at all, because you haven't gained any new information about what you're trying to measure.)

So, to get good results, you have to let the MCMC simulation wander around parameter space for a while, for a long enough time that the point it's at is no longer closely related to the point it was at before. (Move one small step away from a point and you're guaranteed to be close to that point; make many random steps, and you could be anywhere, near or far.)

One way to guess whether you've moved enough is to pick some observable (a function of the system state), and see how closely correlated it is with the value of that observable at a previous state. You typically find that the correlation of an observable with its initial value decreases exponentially with the number of steps made, which suggests that the state of the system itself decorrelates exponentially with the number of MCMC moves. The characteristic decay time of that exponential is the autocorrelation time. So, if you let your system wander around for a few system autocorrelation times between measuring samples, the measurements have a good chance of being statistically independent of each other.

3. Dec 9, 2003

### Moetasim

Thank you very much. It almost suffices my requirement.