**1. Introduction**

The quality of higher learning and research in a university is usually measured by prestige or by publicity, even opinion surveys directed at certain audiences are frequent: general [1], academic [2] or alumni and students [3]. These surveys show results that are consistent with each other, and on many occasions are based on the prestige of an institution, current advertising on social networks and classic mass media (radio, television and print). For this reason, it is common for them to have a bias from a governmen<sup>t</sup> or a university that is using those studies to self-advertise. An ideal characteristic to avoid this type of bias is the use of unsupervised classification systems, which allow finding the "natural" groups of a set of items to classify them according to their inherent properties, because the groups formed are due to the closeness in their attributes.

Gutiérrez, M.Á.; de-los-Cobos, S.G.; Rincón, E.A.; Mora, R.A. A Comparative Ranking Model amongMexican Universities Using Pattern Recognition. *Mathematics* **2021**, *9*, 1615. https://doi.org/10.3390/ math9141615

**Citation:** Urueta, D.E.; Lara, P.;

Academic Editor: Jorge de Andres Sanchez

Received: 6 June 2021 Accepted: 5 July 2021 Published: 8 July 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

All Mexican universities have characteristics which are comparable to each other, such as: full-time professors, number of members in the National System of Researchers (SNI) and number of articles published in journals in different international indexes, among others. In this article, a classification of the 60 largest universities in Mexico was made using the obtained information from the comparative study of Mexican universities carried out by the National Autonomous University of Mexico (UNAM), which is based on the collection, organization and analysis of information obtained from official sources and recognized databases (SEP, CONACYT, INDAUTOR, IMPI, WoS, Scopus, among others). This database is available at www.execum.unam.mx (accessed on 8 February 2019). This information was divided into two groups: higher learning and research. In the first one registered students (technical professional, bachelor's degree, specialty, master's degree and doctorate) were taken into account, as well as the study degree and type of contract that professors have. On the research side, papers in different indexed journals (SCI, Scopus, CONACYT journal quality index) and patents were also considered. Despite the fact that there is a work related to this database in literature [4], it is limited to analyzing a single year. For the present study the available data was used ranging from 2009 to 2017, this range allows to analyze what is the tendency of the Mexican university system, observing the transitions of some universities among different groups over the years.

In this study three well-known classification techniques: *k*-means, Gaussian mixture method (GMM), and spectral clustering were used to analyze the database. Likewise, principal component analysis (PCA) was used, which is a fast and flexible unsupervised method for reducing dimensionality in data [5].

Just as there are different opinion polls, where some emphasize which are the best elements, some others highlight which are the bad elements, in the same way the classifying algorithms will emphasize either the good or bad characteristics.

This article is divided as follows: Section 1 describes the considered classification techniques in this paper. In Section 2 the database and its attributes used are described. Section 3 describes the proposed matrix model (higher learning and research axis and generated sectors). Section 4 includes the application of the model to the case of 60 Mexican universities. Section 5 shows the results obtained and finally conclusions are presented.

#### **2. Technical Classification by Clustering**

Clustering algorithms are methods that divide a set of data into groups in such a way that members of the same group are more similar to each other than members of different groups [6].
