Numerical analysis

To investigate properties of the estimator produced by MIS-WMC algorithm, the starting distribution was chosen to be U(−10, 10) and the target was set to be a mixture of standard distributions as in the one-dimensional example of Ÿ4.1.1, this set-up is visualised in Figure 8.4.

Figure 8.4: Starting and target densities for the MIS-WMC numerical analysis.

The starting density f(·) was chosen such that it covers the target density g(·) and is similar in location. The main idea for doing this is that intermediate densities ftk(·)

will be covering the high density areas of g(·) and produced samples from these intermediate densities will be of more value. If f(·) is chosen to be signicantly dierent in location from the target g(·), then a signicant amount of time is required for a starting distribution to transform into something of a similar shape and location as the target. This would imply that samples produced from intermediate distributions that are closer to the starting one will be of substantially lesser value. In this particular example, after MIS-WMC is performed we end with a sample of size N = 500 from the target g(·) and with 2276 intermediate distributions between t = 1 and t = 1 each with 499 samples. Given that throughout 500 WMC runs, 2276 checkpoints were created it is important to investigate the dierence between samples. The reason for doing this is, if there is a cluster of checkpoints,

an intermediate point will survive through all of them and will be assigned to each intermediate distribution of each of those corresponding checkpoints. This means that majority of samples from intermediate distributions are sharing the same sample points, making an eective sample size smaller.

We examine the sample similarity by constructing a percentage based index Sβ(α),

α, β ∈ (0, 1)and α ≤ β that measures what percentages of samples of fαdistribution

is identical to those of fβ.

Sβ(α) =

γ(tk = β, tl= α)

N − 1 × 100%, (8.1.9) where γ(tk, tl)is the function which returns the number of duplicate samples between

ftk and ftl, with tk ≥ tl. Fixing β, we can investigate how similarity between

samples changes as we keep reducing α to 0. As we can see (Figure 8.5), the sample similarity percentage decays quite slowly with each `lag', which means that given any distribution ftk and two neighbouring distributions ftk−1 and ftk+1, the

samples associated with each of those distributions are almost the same, with only few sample points being unique for each distribution. This observation suggests that it is relevant to include thinning options before using all intermediate distributions in the computation of statistics using MIS-WMC.

Furthermore, we can see that approximately 15% of samples from a starting distribution ended up surviving to the target, i.e. no intermediate jumps were required in those cases to generate a sample from the target. The eciency of the WMC run on N points could be also judged on the amount of starting points that were not required to do any intermediate jumps. For this reason, it is important to pick the best possible starting distribution which would be similar in shape and location.

Figure 8.6 presents samples from couple of intermediate distributions that were created using the checkpoint procedure. Not ignoring the intermediate points

produced by WMC, we ended up with 2276 intermediate distributions each containing 999 samples, which could be used for statistic computation purposes of the target distribution.

To analyse distribution properties of the MIS-WMC mean estimator Mw,

Mw(λ) = 1 G r X k=1 1 Nk Nk X n=1 wk(xn,k; λ)xn,k g(xn,k) fk(xn,k) , (8.1.10) we will focus on using the time threshold weight function wk(x; λ). To explore the

variance of the estimator, we will set up a simulation that will run 100 times for N = {10, 25, 100, 1000}and for each of N we will record the Mw(λ) estimator value

Figure 8.5: Each trace indicates at what Time (intermediate distribution) what the present sample similarity value is. For example, f1 trace indicates that approximately

for λ = {0.8, 0.9, 0.95}. The goal of the simulation is to spot what eect the λ value together with N has on the variance of the estimator. As expected the empirical mean of the estimator Mw(λ) converges to the true value of the target distribution.

For λ = 0.95 the mean of the estimator seems to be the most consistent around the true value, however it has the highest variance for N = 10 but lowest one for N = 25.

As it could be seen in Figure 8.7, as value of N increases, the benets of using MIS-

Figure 8.6: By not discarding intermediate samples and using the intermediate sample allocation procedure described in Ÿ8.1.5 we are able to produce samples from intermediate distributions. The gure presents histograms of samples from the starting distribution, two intermediate distributions and the target based on N = 1000 points from a starting distribution.

Figure 8.7: Empirical mean of the estimators is plotted after 100 simulation runs together with error bars. Dashed line indicates the actual mean of the target distribution.

WMC are not that clear, however MIS-WMC is doing arguably better than a default mean of the pure WMC for lower values of N. For the small price of the increase in variance, MIS-WMC should be utilised in the situations when computational cost of producing samples from the target is high. In those cases, intermediate samples could play a big role of producing better estimators at almost no additional cost.