Dimension effects

To see the effect of dimension on the accuracy of $\hat{\mbox{\boldmath $\lambda$}}({\bf z}^*)$ as predictor of the asymptotic mean of MCMC-UMS, we repeated the experiment of Figure 5.12 for various manifold dimensions $N-D$. In Figure 5.13, we plotted $\epsilon$ after 20,000 full iterations of MCMC-UMS (systematic approach and whitening). Plotted is the average of $\epsilon$ for three trials at each dimension. We see that even for manifold dimension 3, there is good accuracy. Note that $\epsilon=.003$ means that the fractional error has a standard deviation of .055 or 5.5% error. At a dimension of 125, $\epsilon$ is about .00009, a fractional error standard deviation of 1%.

Figure 5.13: Mean square fractional error as a function of dimension estimated using 20,000 iterations.