3.2.3. Simulation 2 Results

In Figure 6, we show the clustering results from MCLUST-ME and MCLUST (*G* = 2). On this data set, the hard partitioning results do not differ much between the two methods: only two points were classified differently by the two methods (highlighted by black circles).

**Figure 6.** Clustering results for Simulation 2. The clustering results are indicated by different colors and symbols. Points with crosses are misclassified points. The two points that are classified differently by MCLUST-ME and MCLUST are circled in black.

Our focus here is on comparing the classification uncertainties estimated under the two methods. For MCLUST, the uncertainty measure for a point depends only on the point location and estimated centers and covariance matrices of the two clusters. Under MCLUST-ME, the uncertainty measure will also depend on the error covariance associated with the point. When two points are at the same location, MCLUST-ME will give higher uncertainty estimate to the point with greater error covariances (see Equation (12)), which is reasonable. In Figure 7, for each observation, we visualize the change in estimated membership probability to cluster 1 between MCLUST and MCLUST-ME with respect to the magnitude of its error covariance(*Si*): the closer the membership probability is to 0.5 the higher the classification uncertainty. The points with most changes in estimated membership probabilities are highlighted in Figure 8. Relative to MCLUST, the MCLUST-ME model tends to adjust the classification uncertainties upwards for points with high error covariances and downwards for points with low error covariances. In other words, relative to the MCLUST-ME results, MCLUST tends to overestimate clustering uncertainties for points with low error covariances and underestimate clustering uncertainties for points with high error covariances. This is expected, as MCLUST treats all points as measured with no errors and absorbs all individual measurement/estimation errors into the variance estimates for the two clusters. As a crude approximation, one can think that MCLUST effectively treats each point as having an error covariance matrix close to the average of all true error covariances. However, the up or down changes in membership probabilities (and thus uncertainty estimates) are not a simple function of *Si*, and we do not see a clear-cut boundary between the ups and downs in Figure 7, as the estimates of membership probabilities are also affected by differences in estimates of centers and covariance matrices of the two clusters.

**Figure 7.** Change in estimated membership probability to cluster 1 from MCLUST to MCLUST-ME, plotted against error magnitude. *X-axis*: magnitude of error covariance, *Si*; *Y-axis*: estimated membership probabilities to cluster 1 by MCLUST (black dots) and by MCLUST-ME (arrowheads). Changes in estimated membership probabilities from MCLUST to MCLUST-ME are highlighted by arrows (no arrow indicates a change less than 0.01). Blue and red arrows indicate an increase and decrease in estimated clustering uncertainty, respectively. With two clusters, the closer the estimated membership probability to 0.5, the higher the classification uncertainty; the classification membership changes when an arrow crosses the horizontal line at 0.5 (the dashed line).

**Figure 8.** Points with most changes in estimated membership probabilities to cluster 1 from MCLUST to MCLUST-ME. The colored dots correspond to points with a change greater than 0.1 in estimated membership probability to cluster 1. Blue and red colors indicate an increase and decrease in estimated clustering uncertainty, respectively.
