**5. Conclusions**

The analysis of the data with *k*-means shows congruen<sup>t</sup> results year after year, this is demonstrated with the institutions of excellence that remain unchanged, such as: National Polytechnic Institute (IPN), the Metropolitan Autonomous University (UAM) and the University of Guadalajara (UdeG). The group of excellence was joined by the Autonomous University of Nuevo León (UANL) in 2011, the Monterrey Institute of Technology and Higher Education (ITESM) in 2014, the Meritorious Autonomous University of Puebla (BUAP) in 2015 and the Autonomous University of Mexico State (UAMEX) in 2017.

The *k*-means results also indicate that the IPN Center for Research and Advanced Studies (CINVESTAV) remains unchanged within the group of consolidated research institutions; BUAP enters this group, although intermittently in 2011 and 2014; as well as the Autonomous University of San Luis Potosí (UASLP) in the years 2011, 2013, 2014 and 2017. Also, for the last year analyzed, the Iberoamerican University (IBERO) enters the group.

Regarding to the group of institutions consolidated in higher learning, the invariable members are: The National Pedagogical University (UPN) and the University of the Mexican Valley (UVM). The Autonomous University of Baja California (UABC), La Salle University (LASALLE), the Technological University of Mexico (UNITEC) and the Autonomous University of Sinaloa (UAS) were added to it in 2016.

As with *k*-means, the results with GMM also show to be consistent from year to year. for example, in the institutions of excellence that remain unchanged there are two: the IPN and the UAM. This group is joined by the UdeG that occupies this position from 2009 to 2012 and from 2016 to 2017; the UANL which appears in the years 2010 to 2012 and in 2016 and 2017; as well as the ITESM, the BUAP and the UAMEX who enter during the year 2016.

Regarding the group of consolidated research institutions, CINVESTAV is an invariable member; IBERO and UASLP, which entered in the last year of analysis; while the Autonomous University of Morelos State (UAMOR), the University of Guanajuato (UGTO) and the UV only occupy this place from 2010 to 2012 and do not occupy it again in subsequent years.

In contrast, the consolidated institutions in higher learning that remain unchanged are: the UPN and the UVM and although there are also other institutions in the group, they have occupied this place only in certain periods; Such is the case of the Autonomous University of Sinaloa (UAS) in conjunction with LASALLE and the UABC, which are integrated in the period from 2014 to 2016; BUAP and UAMEX only in 2013 and UNITEC with UP, who only joined in 2016.

Just as *k*-means and Gauss highlight which are the best institutions, the spectral grouping highlights which are the institutions with the lowest levels of higher learning and research. That is why, despite showing grouped results in a different way, this technique also shows congruen<sup>t</sup> results in the same way as *k*-means and GMM.

For spectral grouping, in the institutions with good higher learning results, but few research results, there are La Salle University (LASALLE), the Autonomous University Benito Juarez of Oaxaca (UABJO), and the Autonomous University of Guadalajara (UAG), the Autonomous University of Coahuila (UAdeC), the Autonomous University of Chiapas (UANCH), the Technological University of Mexico (UNITEC), the UP and the UVM as permanent members throughout the analyzed period; other institutions such as the Technological Institute of Sonora (ITSON), the Autonomous University of Aguascalientes (UAA), the Autonomous University of Chihuahua (UACH), the University of the Mexican Army and Air force (UDEFA), the Juarez Autonomous University of Tabasco (UJAT), Juarez University of Durango State (UJED), Autonomous University of Nayarit (UAN), Popular Autonomous University of Puebla State (UPAEP), Autonomous University of Tlaxcala (UATX), University of Monterrey (UDEM) they have a consistent presence, although not continuous within the group, in addition to these there are others that due to their small number of appearances within the group were not mentioned.

In the case of institutions with less higher learning and more research, there is none that appears uninterruptedly within the group, because in 2016, the group was empty; However, the College of Postgraduates (COLPOS) and the Autonomous Technological Institute of Mexico (ITAM), are constant members of this group except for the mentioned year; other institutions that also appear frequently within this group are the College of México (COLMEX), the Chapingo University (CHAPINGO), the Autonomous University of Mexico City (UACM) and the University of the Americas Puebla (UDLAP).

Finally, in the least prominent group in higher learning and research, there are the Autonomous Agrarian University Antonio Narro (UAAAN), the Autonomous University of Baja California Sur (UABCS), the Regiomontana University (UERRE), the Autonomous University del Carmen (UNACAR), the International University (UNINTER) and the University of Quintana Roo (UQROO) as permanent members, on the other hand, CHAPINGO, Western Institute of Technology and Higher Studies (ITESO), the UATX, although they are not permanent participants, they appear with some frequency; that is, they have more than four appearances in the group.

The results found by *k*-means and GMM show the constant effort of the institutions to consolidate themselves in higher learning, research or being of excellence; Despite the fact that *k*-means and GMM are techniques with different approaches, both highlight the best institutions, this is the reason why the results obtained between both are consistent; for example, in the case of institutions of excellence, the IPN and the UAM are constant in both analyzes, as is the CINVESTAV for consolidated research institutions. As for the consolidated institutions in higher learning and static, the universities that remain unchanged are exactly the same in both techniques, these are: UVM and UPN for the first group and CINVESTAV for the second.

The results provided by the spectral grouping emphasize the institutions and their constant effort to improve or maintain their status, whether in aspects of higher learning, research or both subject to the resources they possess.

In contrast to all the results shown, it must be said that the National Autonomous University of Mexico (UNAM) is in all years the best institution in both higher learning and research.

**Author Contributions:** Conceptualization, S.G.d.-l.-C.; Formal analysis, E.A.R.; Funding acquisition, R.A.M.; Investigation, D.E.U.; Project administration, M. Á.G.; Software, D.E.U.; Supervision, P.L.; Validation, D.E.U. and S.G.d.-l.-C.; Writing—original draft, D.E.U.; Writing—review & editing, P.L., M. Á.G. and E.A.R. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Estudio Comparativo de las Universidades Mexicanas—Explorador de datos (ExECUM). Is available at http://www.execum.unam.mx/ (accessed on 5 January 2020). ExECUM 2009–2017. Is available at https://github.com/crownirv/execum (accessed on 2 July 2021).

**Conflicts of Interest:** The authors declare no conflict of interest.
