Statistical inefficiency

In this section we present in some detailed form the statistical correlation between the values of a property in a chain of successive values generated by computer simulation. Two concepts are used, the statistical correlation and the statistical inefficiency. Although these two concepts are related they are obtained by using different... 

The second method used to calculate the necessary interval to obtain statistically uncorrelated configurations is the statistical inefficiency. As defined by Friedberg and Cameron [45], it is based on the variance of averages taken over blocks of a chain /). Dividing the chain into Hb blocks with Lb successive values, where L = nb Lb, then the average taken over the bth block is... 

Figure 7-2. Typical graphic of the statistical inefficiency. In this example it is calculated for the case of benzophenone in water simulated with Monte Carlo Metropolis method. The behavior of S(Lb) versus Lb for chains with different sizes L. The dashed line represents the asymptotic value s of uncorrelated blocks as shown in Eq. (7-11)...

In our application, r is therefore an estimate of the statistical inefficiency in the measurements that has routine use in numerical analysis. Later it will be shown that r has a dramatic effect on the results found u.sing Eqs. 11-16, and hence the choice of estimators is an important consideration. To estimate r,-, two different techniques have been employed. A crude estimator can be constructed from the acceptance ratio with number of trial configurations number of accepted configurations... 

Figure 1 Statistical inefficiency from a MC simulation of acetone in water...

Figure 1 shows the statistical inefficiency obtained after a conventional MC simulation of one acetone molecule surrounded by 343 waters. This is a typical figure for the statistical inefficiency s(24). Performing an ensemble average of the solvatochromic shift over all structures generated by the MC simulation is indeed a great waste as in fact only one after around each 800 MC steps is relevant. This dramatic reduction in the number of quantum mechanical calculations needed can be favorably used. 

It is interesting to mention that the statistical inefficiency is obtained only after performing the simulation and therefore it is of little practical use until the simulation is halted. However, for our applications in solvent effects it shows that within a given accuracy determined only by the total number of MC steps there is a great saving of the total calculation if only uncorrelated structures are used in the subsequent quantum mechanical calculations. For the case of the acetone-water calculations only 40 uncorrelated structures are used for the ensemble average of the calculated quantum mechanical results, instead of the total number of 32000. 

M here is the actual number of steps or iterations in the simulation. If the value of s can be reduced, then a more accurate average value can be calculated for a given length of simulation. This should be an important consideration when deciding what simulation protocol to use. For example, it may be more appropriate to use a complex simulation algorithm than a simpler one if the statistical inefficiency is significantly reduced. 

Table 4. Time requirements and statistical inefficiencies, 1(0), for the long runs performed on the Cm polybead and the C, polylxead sterns...

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