5.8 Limitation of outlying energies

It is highly undesirable for an objective function to have a large variance. A larger variance implies that a greater number of configurations is required to determine the objective function to a given accuracy. As noted above, only the variances and not the means of $A(\alpha)$, $B(\alpha)$, and $C(\alpha)$ are significantly affected by the outlying configurations. A decision was made to limit the outlying local energies. An alternative would be to delete the outlying configurations, but this introduces a greater bias and is not as convenient in correlated-sampling schemes. The limiting must be done by the introduction of an arbitrary criterion, which was implemented as follows. First the standard deviation of the sampled local energies, $\sigma$, is calculated. Limiting values for the local energy are then calculated as those beyond which the total expected number of configurations based on a normal distribution is less than $\Delta$, where

$$\Delta = N_s \times 10^{-p}\;\;,$$ (5.15)

$N_s$ is the total number of configurations and $p$ is typically chosen to be 8, although varying $p$ from 4 to 12 makes no significant difference to the results.

A factor of $N_s$ was included, rather than limiting the energies beyond a given number of standard deviations to incorporate the concept that as more configurations are included, the sampling is improved. In the limit of perfect sampling, $N_s\rightarrow\infty$, the objective functions are unchanged. For the model silicon system, the percentage of configurations having their local energies limited by this procedure, with $p$ = 8, is only 0.024% and 0.047% for $\alpha_0$ = 0.03 and 0, respectively. These values corresponds to those configurations lying beyond 5.7 standard deviations from the mean. The effect of limiting the outlying local energies is illustrated in figure 5.4. In figures 5.4a and c the mean values of the objective functions $C$ and $A$ versus $\alpha$ for configurations generated with $\alpha_0$ = 0.03, with values of the limiting power, $p$, in equation 5.15, of 4, 8, 12 and infinity (no limiting) are shown, while in figures 5.4b and d their variances are plotted.

The mean values of $C$ are hardly affected by the limiting, while those of $A$ are only slightly altered. The smaller variances of $C$ and $A$ obtained by limiting the values of the local energy are very clear. If the local energies are not limited then the variances of the objective function are not very accurately determined, even with the large samples of $0.96\times 10^{6}$ configurations. Similar results hold for configurations generated with $\alpha_0$ = 0. This simple procedure can greatly reduce the variance of the variance-like objective functions without significantly affecting their mean values. Limiting the local energies is even more advantageous when small numbers of configurations are used. Limiting the local energies in this way gives significantly better numerical behaviour for all the variance-like objective functions and therefore all data shown in figures 5.5-5.7 have been limited with $p$=8, unless explicitly stated otherwise.

Figure 5.4: The effect of limiting outlying energies on (a,b) objective function $C$ (the unweighted variance) and (c,d) objective function $A$ (the weighted variance) with $\alpha_0$=0.03. Outlying energies are limited as in equation 5.15 with the values of $p$ shown.

Limiting the values of the weights is a crucial part of the variance minimisation procedure for large systems. Comparison of figures 5.4b and d shows that the variance of the unweighted objective function $C(\alpha)$ is smaller than that of the weighted objective function, $A(\alpha)$, for all values of $\alpha$, provided one limits the local energies. The variances close to the minimum are similar but away from the minimum the variance of $A$ increases much more rapidly than that of $C$. The smaller variance of $C$ indicates the superior numerical stability of the unweighted function. Qualitatively similarly behaviour occurs for configurations generated with $\alpha_0$ = 0. A commonly used alternative to setting the weights equal to unity is to limit the maximum value of the weights. In figure 5.5 data for objective function $C$ with the largest value of the weights limited to multiples of 1 and 10 times the mean weight is shown, along with data for the weights set to unity. In this graph the standard deviations of the objective functions are plotted as error bars. Figure 5.5 shows that the variance of $C$ is reduced as the weights are more strongly limited, but the lowest variance is obtained by setting the weights to unity. In addition, when the weights are limited the curvature of the objective function is reduced, which makes it more difficult to locate the minimum. Setting the weights to unity therefore gives the best numerical stability.

Figure 5.5: The unweighted objective function $C$ generated with $\alpha_0=0$ and with limiting of the weights.