## 5 Gaussian Mixture Modeling
### Gaussian Components  
In Table 2, one can see the numerical properties of each Gaussian component considering each of the 30-day snippets that originated it. We sorted each component by ratio average/standard_deviation in ascending order; as a result, the components whose elements have a higher number of hours and higher regularity have a higher rank. This order is fundamental to hour analysis as it carries a semantic characterization of the patients' behavior.  
According to our modeling, refer to Table 2 and Figure 7, ideally, every 30-day period of a patient should be characterized by components 7 (_Ideal_) or 8 (_Plain_), with several hours close to 7 h/night or above 8 h/night and a standard deviation of nearly just 1 h. Component 6 (_Good_) also corresponds to a high adherence, with a time of use above 7 h/night, but with a high standard deviation of \(+/-1.75\) h. Component 5 (_Highly adapted_) indicates periods in which the patient is using the equipment regularly but with room for improvements in the number of hours. Components 4 _Adapted_ is similar to component 5, but with a high instability of \(+/-2\) h. Component 3 _Pre-adaptation_ refers to periods of increasing adaptation, with an average below 5 h/night. However, components 1 _Struggling_ and 2 _Unstable_ refer to periods characterized by failure in using the equipment, with either a very low number of hours or a very high standard deviation, respectively. This classification takes into account the work of [21], who states that a minimum of 4 h/night is necessary for CPAP to be effective; and the work of [45], who verified that the prevalence of sleepiness, around 8.7%, is lower when the CPAP use lasts for more than 6 h/night if compared to patients who use the device for less than 4 h/night, presenting around 18.5% of sleepiness.  
In Figure 7, we plot the eight Gaussian components of our model along with the corresponding instances of each one. In the center of each plot, one can see the mean series (prototype of the Gaussian component) along with the standard deviation and the standard deviation times two. The Gaussian nature is visually plausible.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**\#Comps** & **LL** & **AIC** & **AWE** & **CAIC** & **KIC** & **SABIC** & **IC** \\ \hline \(1\) & \(-331\),566 & 663,173 & 663,594 & 663,355 & 663,196 & 663,271 & \(-663\),335 \\ \hline \(2\) & \(-323\),953 & 647,988 & 648,855 & 648,361 & 648,032 & 648,189 & \(-654\),536 \\ \hline \(3\) & \(-319\),721 & 639,567 & 640,878 & 640,130 & 639,632 & 639,871 & \(-648\),605 \\ \hline \(4\) & \(-318\),231 & 636,627 & 638,383 & 637,381 & 636,713 & 637,035 & \(-649\),319 \\ \hline \(5\) & \(-317\),349 & 634,907 & 637,108 & 635,852 & 635,014 & 635,417 & \(-650\),038 \\ \hline \(6\) & \(-316\),502 & 633,254 & 635,899 & 634,390 & 633,382 & 633,867 & \(-650\),552 \\ \hline \(7\) & \(-315\),985 & 632,261 & 635,351 & 633,588 & 632,410 & 632,978 & \(-650\),243 \\ \hline \(8\) & \(-315\),801 & 631,936 & 635,470 & 633,453 & 632,106 & 632,756 & \(-653\),016 \\ \hline \end{tabular}
\end{table}
Table 1: Additional statistics Log-likelihood (LL), Aikake Information Criterion (AIC), Approximate Weight of Evidence (AWE) [41], Consistent Aikake Information Criterion (CAIC), Kullback Information Criterion (KIC) [42], Sample Size-adjusted Bayesian Information Criterion (SABIC) [43], and Integrated Completed Likelihood (ICL) [44].  
Figure 7: Time series plot of the eight components that characterize the CPAP use 30-days snippets extracted from the CPAP use dataset. Each plot presents the mean (thickest red line), the variance, and two times the variance lines. Notice that we present the average because it is a parameter to the Gaussian distribution; we do not depend on this computation to characterize the time series.