Block averaging

Flere we discuss only briefly the simulation of continuous transitions (see [132. 135] and references therein). Suppose that tire transition is characterized by a non-vanishing order parameter X and a corresponding divergent correlation length We shall be interested in the block average value where the L... 

As the number of steps fj, in each block increases, so it would be expected that the block averages become uncorrelated. When this is the case, then the Vciriance of the block averages, will become inversely proportional to is calculated as follows ... 

Next consider the case that uses randomized blocking to eliminate the effect of some variable whose effect is of no interest, such as the batch-to-batch variation of the catalysts in the chemical reactor example. Suppose there are k treatments and n experiments in each treatment. The results from nk experiments can be arranged as shown in the block design table within each block, the various treatments are applied in a random order. Compute the block average, the treatment average, as well as the grand average as before. 

Number without any parentheses is the size of the block (in ps), number with parentheses is the number of blocks these are determined after the equilibration sections of the trajectories are removed. d In kcal/mol the number in parentheses is the statistical error evaluated based on block average. 

Block averaging comes from a set of N work values W, Wi,. .., Wn generated in a NEW simulation. The free energy difference AA can be estimated by computing the block average with different block sizes n... 

Fig. 6.12. Block averaging of an FEB or NEW calculation can be used to create a smoothed running profile. This contrasts with the jagged steps of a running average based on the same data. The running steps occur with the relatively rare sampling of events in the tails of the distribution that dominate the relative free energy estimate. The smoothed running curve uses the block procedure described in the text...

Fig. 6.13. Extrapolation to a free energy estimate based on block averages can best be analyzed in a 1/N plot. In this type of plot the large-A limit is towards the origin, rather than increasing to the right. By fitting the data to an extrapolating estimate of the expansion (see text) an estimate of the free energy difference can be made...

Keywords error analysis principal component block averaging convergence sampling quality equilibrium ensemble correlation time ergodicity... 

Both the correlation-time analysis and the block-averaging scheme described below assume that a dynamical trajectory is being analyzed. Again, by "dynamical" we only mean that correlations are "transmitted" via sequential configurations — which is not true in a method like replica exchange. 

When executed properly, the block-averaging analysis automatically corrects two of the weaknesses in correlation-time estimates of the error based on... 

It is not uncommon for researchers to use the name "block averaging" to describe a second, far simpler procedure. In this case, a single time series is split into M blocks, and the variance between the averages for those blocks is presented as the uncertainty. However, unlike the true block-averaging protocol described above, this procedure is not statistically meaningful, because the... 

Figure 6 The block-averaging procedure considers a full range of block sizes. The upper panel shows the time series for the squared cosine of the central dihedral of butane, with two different block sizes annotated. The lower panel shows the block-averaged standard error for that times series, as a function of block size.

Single observables. Block averaging is a simple, relatively robust procedure for estimating statistical uncertainty. Visual and correlation analyses should also be performed. 

The square root of the variance, the standard deviation, oy, thus quantifies the width or spread in the distribution it has the same units as / itself, unlike the variance. Except in specialized analyses (such a block averaging) the variance does not quantify error. As an example, the heights of college students can have a broad range — that is, large variance — while the average height can be known with an error much smaller than the standard deviation. 

---