## 6 K-Mean Clustering Analyses Applied to MILs  
In the previous section, we have seen the broad and sometimes skewed distribution of and seasonal variation in the MIL characteristics. We know from Section 1 that there are several mechanisms proposed for the creation of MILs. Directly linking complex, nonlinear, gravity-wave-dissipating phenomena measured by single site Rayleigh lidar systems directly to other quantities such as wave flux is a daunting prospect. Further complications arise, for example, that we do not have simultaneous measurements of the winds and the temperatures and that we cannot work out intrinsic gravity wave parameters or energy dissipation directly. Given these observational constraints, we wanted to try a different approach and use a clustering technique on this MIL climatology to test the concept of sorting MIL observations into self-similar categories. In this trial study, we will start with a simple K-mean clustering technique, and we will choose k = 3. In the future, we might try a more complex technique with more dynamic variable selection. Clustering is a descriptive technique[41] when different populations are suspected; therefore, in this study, the k-means clustering method was used for more detailed MIL characteristics analysis to obtain a 3D model of MIL classifications. Cases in the same cluster are similar with respect to the information, cluster analysis was based on variables inducing the dissimilarities. According to the k-means clustering definition, assigning \(n\) MIL observations to one of the \(k\) clusters is defined by centroids, where \(k\) is chosen before the algorithm starts. Then, the point-to-cluster-centroid distances of all observations to each centroid was computed. We should note that to determine distances, the squared Euclidean distance metric has been used. To proceed, an observation was selected uniformly at random from the data set X (main MILs parameters: amplitude, thickness, and minima altitude). The chosen observation is the first centroid and is denoted \(c_{1}\). The next step is computing distances from each observation to \(c_{1}\). The Distance between \(c_{j}\) and the observation \(m\) was considering as \(d(x_{m}  
thicknesses between Groups 2 and 3 are observed, with slightly thicker MILs than group 1. The seasonal behavior of the MIL occurrence shows a full annual cycle for the three categories, with a maximum in winter and a minimum in summer by a factor of 6. Based on the results of each MIL's group characteristics that are classified in Table 3, an average of 10.9 K amplitude with 3.1 km thicknesses for Lower MILs calculated in the mean minimum altitude at 57.5 km can be mentioned. However, the mean range values of amplitude for the major upper MIL (27.5 K) are very different from the lower MIL (10.9 K) and the minor upper MILs (10.3 K) values. The overall thickness of major upper MIL and minor upper MIL appears somewhat is evident at first glance that the thickness of major upper MIL (4.5 K) is significantly greater than the other two layers. Figure 6 represents the annual evolution of mesospheric inversion layers, showing decadal oscillations. We see local maxima in MIL occurrence in 1981-1983 and 1990-1991 and potentially a minor maxima from 2000-2003 and 2016 to 2019. We speculate that there may be some phase-shifted connection to the solar cycle. The maximum of solar cycle 21 spanned 1976-1979, cycle 22 spanned 1986-1989, cycle 23 spanned 1996-2001, and cycle 24 spanned 2008 to 2014. Previous work [42; 43] has well established the connection between mesospheric temperatures and the solar cycle. Going forward, it would be interesting to know the potential role of the solar cycle in the observed fluctuation in MIL. As an alternative hypothesis, the increase in the number of MIL observations in recent years could be only related to the improved quality of the Rayleigh lidar devices over the years. The variations in the quality of observational sampling of MILs could dominate the time series.