## 3 Results
### Tempered EM: Application to Mixtures of Gaussian
#### 3.2.2 Experimental Results Analysis  
In this section, we analyse the results of EM, _decreasing_ tmp-EM and _oscillating_ tmp-EM over all the simulations. First, an illustration: Figures 5 and 6 depict the final states of EM and _oscillating_ tmp-EM on one typical simulation for each of the three ambiguity level (the three parameter families) starting from the _barycenter_ and _2v1_ initialisation respectively. The simulated data are represented by the green crosses. The initial centroids are in blue. The orange cross represents the estimated centroids positions \(\hat{\mu}_{k}\), and the orange confidence ellipses  
Figure 5: Typical final positioning of the centroids by EM (first row) and tmp-EM with _oscillating_ profile (second row) **when the initialisation is made at the barycenter of all data points** (blue crosses). The three columns represent the three gradually more ambiguous parameter sets. Each figure represents the positions of the estimated centroids after convergence of the EM algorithms (orange cross), with their estimated covariance matrices (orange confidence ellipses). In each simulation, 500 sample points were drawn from the real GMM (small green crosses). In those example, tmp-EM managed to correctly identify the position of the three real centroids. Figure 6: Typical final positioning of the centroids by EM (first row) and tmp-EM with _oscillating_ profile (second row) **when the initialisation is made by selecting two points in the isolated cluster and one in the lower ambiguous cluster** (blue crosses). The three columns represent the three gradually more ambiguous parameter sets. Each figure represents the positions of the estimated centroids after convergence of the EM algorithms (orange cross), with their estimated covariance matrices (orange confidence ellipses). In each simulation, 500 sample points were drawn from the real GMM (small green crosses). In those examples, although EM kept two centroids on the isolated cluster, tmp-EM managed to correctly identify the position of the three real centroids. are visual representations of the estimated covariance matrices \(\hat{\Sigma}_{k}\). In Supplementary Materials, we show step by step the path taken by the estimated parameters of tmp-EM before convergence, providing much more detail on the method's behaviours. These illustrative examples show _oscillating_ tmp-EM better succeeding at clustering recovery than the classical EM. The results over all simulations are aggregated in Table 1, and confirm this observation. Table 1 presents the average and the standard deviation of the relative \(l_{2}\) error on \(\mu_{k}\) of the EMs. For each category, the better result over the three EM is highlighted in bold. The recovery of the true class averages \(\mu_{k}\) is spotlighted as it is the essential success metric for this experiment. More specifically, class 1 and 2, the two leftmost classes, are the hardest to correctly recover and the ones whose estimation is the differentiating criterion between the algorithms. Indeed, as can be seen in Table 1, \(\mu_{3}\) is always well estimated by all methods. Hence, in the following, we discuss the error on \(\mu_{1}\) and \(\mu_{2}\). With the _barycenter_ initialisation, classical EM and _decreasing_ tmp-EM have similar average relative error levels. With classical EM actually being slightly better. However, _oscillating_ tmp-EM is much better than both of them, with error levels smaller by a factor of 10 on parameter families 1 and 2, and by a factor of 5 on parameter family 3. The standard deviation of _oscillating_ tmp-EM is also lower, by a factor of roughly 3 on parameter families 1 and 2, and by a factor of 2 on parameter family 3. With the _2v1_ initialisation, all error levels are higher. This time, _decreasing_ tmp-EM is better in average than classical EM by a factor of 1.7 to 1.4, depending on the parameter family. In turn, _oscillating_ tmp-EM is better than _decreasing_ tmp-EM by a factor of 3.1 to 3.4 depending on the parameter family.