# On the Optimum Linear Soft Fusion of Classifiers
## 5 Real Data Experiments  
In this section, we consider some experiments with real data. This involves the automatic classification of sleep stages based on the information provided by different biosignals (polysommograms). This classification is usually performed manually by a medical expert, resulting in a slow and tedious procedure. In particular, we consider the detection of very-short periods of wakefulness [29; 30] (also called arousals) as their frequency is related to the presence of apnea and epilepsy. The objective is by no means to present the most-precise method for solving this complex problem, among the myriad of options and variants available. Rather, it is to verify in a real-life context how the fusion of several classifiers achieves results consistent with the theory outlined above. The data were obtained from the public database Physionet [31]. Ten patients were considered. Every patient was monitored during sleep, thus obtaining the corresponding polysommogram. In this experiment, we considered the electroencephalograms (EEGs). Then, 8 EEG features were obtained in non-overlapping intervals of 30 sec (epochs) from every EEG recording. The number of epochs varied for every patient, ranging from 749 (6 h and 14 min) to 925 (7 h and 42 min). The 8 features were: Powers in the frequency bands delta (0-4 Hz),  
Figure 5: (**a**) Accuracy after fusion as a function of the accuracy of the best classifier; (**b**) accuracy after fusion as a function of the accuracy of the best classifier. theta (5-7 Hz), alpha (8-12 Hz), sigma (13-15 Hz), and beta (16-30 Hz), and 3 Hjorth parameters: activity, mobility, and complexity. These features are routinely used in the analysis of EEG signals [32; 33]. As we were interested in the detection of arousals, a two-class problem was considered, where class 1 corresponded to arousal and class 0 corresponded to any of the other possible sleep stage. This is in concordance with the two-class scenario assumed in the previous theoretical analysis. Five classifiers were considered: Gaussian Bayes, Gaussian Naive Bayes, Nearest Mean, Linear Discriminant, and Logistic Regression. Table 1 indicates the equations corresponding to the score computation in every method, as well as the estimated parameters during training. The classifiers were separately trained for every patient using the first half of the respective EEG recordings. Then, the second half was used for testing. The scores of all patients were grouped by classes. Thus, a total number of 6706 scores of class 0 and 1708 of class 1 were generated by every classifier. Despite this imbalanced scenario, we assumed that \(P_{0}=P_{1}=0.5\) in all the experiments, in order to not harm the detection of arousals, which was a priority in this particular context. Table 2 shows the samples' estimates of the score bias and variance for every class and every method, according to the definitions of (2) and (3). We can see that class 1 exhibits a significantly higher bias and variance than class 0 in all the five methods. The corresponding _MSE_ and the accuracy are also indicated in Table 2 and used to rank the methods from \"the best\" (top) to the \"worst\" (bottom). The last row of Table 2 shows the statistics corresponding to the scores of the optimal linear combiner (6). Matrix \(M_{s}\) was estimated from the scores of the training set, using sample estimates in (5) for the covariance and bias matrices. Notice that, as expected, the bias of the fused scores does not show a reduction with respect to the bias of the individual classifiers. However, the variance is reduced, this effect being the main reason from the _MSE_ decrease and corresponding accuracy increase achieved by the fusion of classifiers. Certainly, the improvement is modest due in part to the high correlation among the scores of the 5 individual classifiers, as indicated by the correlation coefficient matrices included at the bottom of Table 2. Furthermore, significant