*4.1.* k*-means Results*

This section may be divided by subheadings. It should provide a concise and precise description of the experimental results, their interpretation, as well as the experimental conclusions that can be drawn.

The result of this is that each cluster is associated not with a hard-edged sphere, but with a smooth Gaussian model. Although GMM is categorized as a clustering algorithm, it is technically a generative probabilistic model describing the data distribution; due to this property, there are two important limitations with GMM: the first one is about its computation complexity because it is necessary to calculate the distributions, and whereby the algorithm can fail if the dimensionality of the problem is too high; the second limitation is that in many instances, the number of groups is unknown and it may be necessary to experiment with a number of different groups in order to find the most suitable.

Table 2 shows the classification with *k*-means. Items not included belong to quadrant 1, which includes institutions that are not good in either of the two areas. It can be seen at first glance that there are institutions that remain there throughout time such as The National Polytechnic Institute (IPN), the Metropolitan Autonomous University (UAM) and the University of Guadalajara (UdeG); Likewise, in 2011 the Autonomous University of Nuevo León (UANL) entered this zone and, like the previous institutions, remained there until the last year analyzed. In addition to the UANL, there are two more institutions that follow the same trend, the Monterrey Institute of Technology and Higher Education (ITESM), which has been in this position since 2014, while the Meritorious Autonomous University of Puebla (BUAP) does so from 2015. During the last year of analysis, the Autonomous University of Mexico State (UAMEX) was added to the list. Other institutions can be seen that also vary their position during the analyzed period; these transitions can be observed in Figure 8, where the effort of the institutions to maintain or improve their status is evident throughout the study.


**Figure 8.** Visual representation of the results from 2009 to 2017 applying *k*-means.

4.1.1. Invariable Universities in *k*-Means Analysis

• Excellence

> The invariable universities consolidated in excellence are those that, throughout the analyzed period, that is, from 2009 to 2017, excel in both higher learning and research.

These are: The National Polytechnic Institute (IPN), the Metropolitan Autonomous University (UAM) and the University of Guadalajara (UdeG).


According to *k*-means classification, approximately 70 percent of educational institutions fall into this category (41 of them). Their names are shown in the lower left of Figure 6.

4.1.2. Universities in Transition in *k*-Means Analysis

• Universities that have improved in higher learning:

In 2016 La Salle University (LASALLE), the Autonomous University of Baja California (UABC) and the Technological University of Mexico (UNITEC) became part of the consolidated universities in higher learning.

The Autonomous University of Sinaloa (UAS) shows a tendency to consolidate higher learning because it begins hovering between static universities at the beginning of the study, and although for 2012 it is consolidated into higher learning, in 2014 it returns to the place where it started; finally, in 2016 it was consolidated again in higher learning and remained on the site until the last year analyzed.

• Universities that have improved in research:

During the first years of analysis, though the Autonomous University of San Luis Potosí (UASLP) is a static institution, in 2011 it was consolidated in research, the following year it was once again part of the static ones; and this behavior is repeated in the following two years until it remains from 2014 to 2016 as a static institution and during the last year analyzed it is consolidated again in research. This behavior of constant transitions makes evident their interest in consolidating in research.

The Iberoamerican University (IBERO) remains among the static institutions for almost the entire period analyzed and it is not until the last year that this institution is consolidated in research.

• Universities that became of excellence:

> There are institutions such as the Autonomous University of Nuevo León (UANL) and the Monterrey Institute of Technology and Higher Education (ITESM) which begin the study as consolidated in higher learning, the study shows their interest in being of excellence, achieving their goal for the year 2011 and 2016, respectively.

> The Meritorious Autonomous University of Puebla (BUAP) went from being a static university to one with a better research quality in 2011; However, by 2012 it was once again part of the static institutions, where it remained until 2013. During 2014 its research quality improved again and for the following year it was able to be part of the excellence institutions, maintaining that position until the last year of analysis.

> The Autonomous University of Mexico State (UAMEX) is another institution that begins located among the static universities, but in 2012 it became consolidated in higher learning and only became static again during 2016, because for the following year it passes to be part of the excellence group.
