A Comparative Ranking Model among Mexican Universities Using Pattern Recognition

: The evaluation of quality in higher education is today a matter of great importance in most countries because the allocation of resources should be in accordance with the quality of universities. Due to this, there are numerous initiatives to create instruments and evaluation tools that can offer a quality comparison among institutions and countries, the results of these efforts used to be called international rankings. These rankings include some that are “reputational” or subjective, based on opinion polls applied to groups that, which is estimated, can issue authorized views. There are also “objective” rankings, based on performance indicators, which are calculated from a certain set of empirical data; however, on many occasions these indicators are sponsored by universities with the desire to appear among the best universities and emphasize some characteristics more than others, which makes them untrustworthy and very variable between each other. In this sense, we considered the Comparative Study of Mexican Universities (CSMU), a database of statistical information on education and research of Mexican higher education institutions, this database allows users to be responsible for establishing comparisons and relationships that may exist among existing information items, or building indicators based on their own needs and analysis perspectives (M á rquez, 2010). This work develops an unsupervised alternative model of ranking among universities using pattern recognition, speciﬁcally clustering techniques, which are based on public access data. The results of the CSMU database are obtained by analyzing 60 universities as a ﬁrst iteration, but to present the ﬁnal results UNAM is excluded.


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 government 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. The goal of k-means is to minimize the sum of the squared error over all groups denoted by J(C); Thus, the objective function is stated as: (1) Minimizing this objective function is an NP-Hard problem, even for k = 2 [8]. Therefore, k-means, is a greedy algorithm, this means, that it builds up a solution choosing the best option at every step so it can be expected to converge to a local minimum. k-means starts with an initial partition with k groups and assigns observations to groups to reduce the squared error. Since the squared error tends to decrease with an increase in the number of k groups (with J(C) = 0 when k = n), it is minimized for a fixed number of groups [9].
A k-means algorithm requires some user-specified parameters such as the number of clusters: typically, k-means runs independently for different values of k and the partition that appears the most meaningful to the human expert is selected. Besides, different initializations can lead to different final clusters because k-means can only converge to local minima. Another user-specified parameter is the metric, while it is true that the most used metric for computing the distance between points and cluster centers is the Euclidean distance, which is why k-means is limited to linear cluster boundaries; however, some other metrics such as the Mahalanobis distance metric has been used to detect hyperellipsoidal clusters [10]. Moreover, it has a limitation about the number of observations, because k-means assumes that each group has roughly the same cardinality.
Due in k-means there is no assurance that it will lead to the global best solution, k-means run for multiple starting guesses, and it improves the result in each step. However, k-means is used because it is broadly easy and fast to code and implement it.

Gaussian Mixture Model
A Gaussian mixture model is a probabilistic model that assumes all the data points are generated from a mixture of a finite number of Gaussian distributions with unknown parameters. A GMM can be seen as a k-means generalization which incorporates information about the covariance structure of the data, as well as the centers of the latent Gaussians [11]. GMM attempts to find a mixture of multi-dimensional Gaussian probability distributions that best model any input dataset. The pseudocode is: Begin 1. Choose starting guesses for the location and shape 2. While the convergence is not reached: For each point, find weights encoding the probability of membership in each cluster. For each cluster, update its location, normalization, and shape based on all data points, making use of the weights.

End While End
Formally, let k and n be the number of clusters and the total number of observations, respectively. Let µ c , Σ c and π c be the mean, covariance, and the mixing probability of cluster c, 1 ≤ c ≤ k. For GMM, µ c is the center of cluster c, Σ c which represents its width and π c defines how large or small the Gaussian function will be.
Then the probability that x i , 1 ≤ i ≤ n is in the cluster c is given by: where N (x | µ, Σ) describes the multivariable Gaussian. γ c i gives the probability that x i is in cluster c, divided by the sum of the probabilities that x i is in cluster c , for all 1 ≤ c ≤ k, so if x i is very close to a Gaussian c, it will have high values of γ c i and relatively low values for any other case. As a second step, for each cluster c: the total weight m c is calculated (which can be considered as the fraction of points assigned to group c) and π c , µ c and Σ c are updated using γ c i with: Finally, the first and second steps are repeated until convergence is reached [12]. 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 distribution of the data; 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.

Spectral Clustering
Spectral clustering is a technique whose goal is to cluster data that is connected, but not necessarily clustered within convex boundaries, so it has no limitations on the shape of data and can detect linearly non-separable patterns. The basic idea is to construct a weighted graph from the initial dataset where each node represents a pattern, and each weighted edge simply considers the similarity between two patterns [13]. In this context, this clustering problem can be seen as a graph cut problem, which can be tackled by means of the spectral graph theory. The core of this theory is the eigenvalue decomposition of the Laplacian matrix of the weighted graph obtained from data. The pseudocode is: Begin 1. Compute A, the n × n affinity matrix 2. Get the eigensystem of A: Compute the first k eigenvectors of its Laplacian matrix to define a feature vector for each object: Set U = n × k matrix containing the normalized eigenvectors of the k largest eigenvalues of A in its columns 3. Apply k-means on the row space of U to find the k cluster End Formally, let n be the number of data points to be grouped and W = w i, j nxn the weight matrix where each w i, j is the similarity between x i and x j data points. So, a clustering problem can be formulated into the minimum cut problem, i.e., where q = (q 1 , q 2 , . . . , q n ) is a vector for binary memberships and if we express a partition (A, B) as the vector q i , each q i can be 1 if i ∈ A or −1 i ∈ B. L is the Laplacian matrix, defined as For grouping into several classes, the objective function can be defined as: where k is the number of clusters, q ∈ {1, 2, . . . , k} n , C z, z , = ∑ n i, j=1 δ(q i , z)δ q j , z , w i, j and D z = ∑ n i=1 ∑ n j=1 δ(q i , z)w i, j . However, efficiently finding the solution that minimizes the above equation is quite difficult. Therefore, a common strategy is to first get the smallest v eigen-vectors of the Laplacian matrix L (excluding the one with zero eigen-value), and project the data points in the low-dimensional space spanning the v eigen-vectors. Then, a standard clustering algorithm, such as k-means, is applied to the cluster data points in this low-dimensional space [14].
In short terms, spectral clustering is based on two main steps: first embedding the data points in a space in which clusters are more "obvious" (using the eigenvectors of a Gram matrix), and then applying a classical clustering algorithm such as k-means [15]. The affinity matrix M is formed using a kernel such as the Gaussian kernel. To obtain m clusters, the first m principal eigenvectors of M are computed, and k-means is applied on the unit-norm coordinates. So, if we consider a data set which consists of n data points, the time complexity of spectral clustering is O n 3 , which makes it prohibitive for large-scale data application [16]. Moreover, there is evidence [13] that spectral clustering can be quite sensitive to changes in the similarity graph so noisy datasets can cause problems.
Choosing the number k of clusters is a general problem for all clustering algorithms, and just like k-means and GMM, it requires the number of clusters to be specified.

Principal Component Analysis
Principal component analysis (PCA) is a linear dimensionality reduction technique that can be used to extract information from a high-dimensional space by projecting it onto a lower-dimensional subspace. It tries to preserve the essential parts that have more variation of the data and eliminate the non-essential parts with less variation. It does this through a statistical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables (entities each of which takes on several numerical values) into a set of linearly uncorrelated variable values called principal components. In short, what the algorithm does is [5] Thanks to the PCA you can get: • A measure of how each variable is associated with the others (covariance matrix) • The direction in which our data is scattered (eigenvectors) • The relative importance of these different directions (eigenvalues).
In summary, PCA dimensionality reduction causes the least important attribute information to be removed, leaving only the data components with the highest variance, that is, the resulting data retains the maximum data variance. For this reason, although PCA is used to reduce the dimensionality in the data, may also be useful as a visualization tool, for filtering noise and for feature extraction.

Determine the Number of Clusters and Evaluate Clustering Performance: Silhouette Coefficient
Silhouette Coefficient or silhouette score is a cluster validity measure for evaluating clustering performance. To calculate the Silhouette score for each observation/data point, the following distances need to be found out for each observation belonging to all the clusters.

•
Mean distance between the observation and all other data points in the same cluster. This distance can also be called a mean intra-cluster distance. This mean distance is denoted by a(i) • Mean distance between the observation and all other data points of the next nearest cluster. This distance can also be called a mean nearest-cluster distance. The mean distance is denoted by b(i) Silhouette score, s(i), for each sample is calculated using the following formula: Silhouette coefficient values ranges from −1 to 1. Silhouette coefficients near +1 indicate that clusters are well apart from each other and clearly distinguished. A value of 0 indicates that the clusters are indifferent, or we can say that the distance between clusters is not significant and negative values indicate that clustering configuration may have too many or too few clusters. Since silhouette coefficients are used to study the separation distance between the resulting clusters it is possible to use it to select the number of clusters for clustering techniques.

Comparative Study of Mexican Universities
The Comparative Study of Mexican Universities [17] is a research project developed by the General Directorate of Institutional Evaluation of the National Autonomous University of Mexico (UNAM) that systematizes, analyzes, and disseminates statistical series, compiled in official sources and recognized databases, which allow to contrast the development of Mexican universities in their substantive functions: higher learning, research, and dissemination of culture.
The CSMU is not a hierarchical classification (ranking) of Mexican higher education institutions but rather it is presented as an alternative to the existing rankings; because its objective is not to rate the universities or build regulations under certain assumptions about the quality or prestige of the institutions and their programs, in contrast, it seeks to provide items of information from public access sources, objective data that covers both the characteristics of institutions such as the substantive functions of university activities.
In this sense, the CSMU favors the presentation of raw data without the use of groupings or weightings, because this type of practice causes the results to always end up being questioned. These characteristics of the CSMU allow users to be responsible for establishing the comparisons and relationships that may exist among the different existing information items, or building indicators based on their own needs and analysis perspectives. Likewise, users are responsible for adapting their interpretations to the different characteristics that Mexican universities have among them [18].
The CSMU data in this study include 60 Mexican universities (45 public and 15 private) but the UNAM from 2009 to 2017. These universities concentrate more than 50 percent of Mexico's higher education enrollment. The database provides information on the following items: • Teachers, tuition, and academic programs.  The results of the study for each of these nine items are published on a dynamic web page with systematized information, which can be consulted through the Data Explorer of the Comparative Study of Mexican Universities.

Application Instance: 60 ExECUM Universities
The ExECUM database was split into two independent databases taking into account the factors of higher learning and research. The information contained into the higher learning independent database is as follows: Teachers instructing

Proposed Matrix Model
Among many activities that take place within a university (management, dissemination of culture, sports activities, among others), the most important areas were the training of undergraduate and graduate students, as well as research. Only these two last items were taken into consideration for the proposed model. The available data referring to higher learning were used, such as: number of full-time or part-time teachers, maximum degree of studies, number of enrolled students, number of graduated students, and academic programs offered. While in research part, the number of research articles that are in different international indexes (JCR, ISI, Scopus, Latindex, Zentralblat Math, among others) can be considered, as well as the number of patents generated or citations in international journals.
Universities can be classified using the clustering strategies previously described and historical data. From available data it is possible to assign an order from highest to lowest; for example, considering the distance of the centroids with respect to the origin, the centroids closest to the origin imply a lower performance (fewer graduate students or fewer research articles generated).
Considering the dimensions already described, a matrix can be structured where the classification according to higher learning can be shown in the vertical axis and research in the horizontal axis, see Figure 1.

Proposed Matrix Model.
Among many activities that take place within a university (managemen tion of culture, sports activities, among others), the most important areas wer of undergraduate and graduate students, as well as research. Only these t were taken into consideration for the proposed model. The available data higher learning were used, such as: number of full-time or part-time teache degree of studies, number of enrolled students, number of graduated stude demic programs offered. While in research part, the number of research ar in different international indexes (JCR, ISI, Scopus, Latindex, Zentralblat M others) can be considered, as well as the number of patents generated or cita national journals.
Universities can be classified using the clustering strategies previously d historical data. From available data it is possible to assign an order from high for example, considering the distance of the centroids with respect to the or troids closest to the origin imply a lower performance (fewer graduate stud research articles generated).
Considering the dimensions already described, a matrix can be structur classification according to higher learning can be shown in the vertical axis in the horizontal axis, see Figure 1. This model is divided into four classification quadrants: the first quad tain static institutions, that is, with minor higher learning and minor research quadrant will have consolidated institutions in higher learning; that is, thos with minor research and major higher learning. The third one will house research institutions, that is, with major research and minor higher learnin the fourth quadrant will be the excellence institutions, this means that thos on this site have the best results in both higher learning and research.
As mentioned above, and in order to locate the institutions, the original This model is divided into four classification quadrants: the first quadrant will contain static institutions, that is, with minor higher learning and minor research. The second quadrant will have consolidated institutions in higher learning; that is, those institutions with minor research and major higher learning. The third one will house consolidated research institutions, that is, with major research and minor higher learning. Finally, in the fourth quadrant will be the excellence institutions, this means that those universities on this site have the best results in both higher learning and research. As mentioned above, and in order to locate the institutions, the original database was divided in two parts: part 1 corresponding to higher learning and part 2 corresponding to research. Each part was solved separately using the aforementioned clustering algorithms. In this way, the cases in Table 1 will be had for each clustering technique: Using the matrix in Figure 1, arrows will be used to show existing the institutions transitions among the quadrants, indicating at the top of each one the year in which they occurred; on the other hand, the highlighted institutions will be those that remained in the same group throughout the study.
Regarding evaluation, two different types of results were obtained: those that include the UNAM and those that do not include it, its presence represents an imbalance for the instances since this institution is quite far from the others in terms of size and, hence, in their higher learning and research capacity, whereby the distances between this institution and the others are shortened.
To demonstrate the above, first PCA analysis was applied on the databases from higher studies and research, and then, it was analysed how many dimensions are necessary to maintain the largest possible variance of both databases. The results of the PCA analysis based on higher studies database from the years 2009 to 2017 are shown in Figure 2:  Using the matrix in Figure 1, arrows will be used to show existing the institutions transitions among the quadrants, indicating at the top of each one the year in which they occurred; on the other hand, the highlighted institutions will be those that remained in the same group throughout the study.
Regarding evaluation, two different types of results were obtained: those that include the UNAM and those that do not include it, its presence represents an imbalance for the instances since this institution is quite far from the others in terms of size and, hence, in their higher learning and research capacity, whereby the distances between this institution and the others are shortened.
To demonstrate the above, first PCA analysis was applied on the databases from higher studies and research, and then, it was analysed how many dimensions are necessary to maintain the largest possible variance of both databases. The results of the PCA analysis based on higher studies database from the years 2009 to 2017 are shown in Figure 2: As it can be seen on Figure 2, only one component represents 85% of variance of the higher studies database. Similarly, an analysis using PCA considering research database is shown in Figure 3:  As it can be seen on Figure 2, only one component represents 85% of variance of the higher studies database. Similarly, an analysis using PCA considering research database is shown in Figure 3: In Figure 3 is shown that only one component represents around 81% of data variance of the research database. After PCA analysis and considering results in Figures 2 and 3, the graph that can be seen in Figure 4 was created; it might seem remarkable that the sum of the variation in this graph exceeds 100%. This is because, as commented in previous paragraphs, the CSMU database was separated into two databases corresponding to higher learning and research and then, reducing all the higher learning database dimensions to a single principal component that is projected on the ordered axis and reducing all the research database dimensions to a principal component which is projected on the abscissa axis; the graph was created maintaining a total data variance of 85% and 81% for every database respectively, for this reason the sum of both axes exceeds 100%. As it can be seen in Figure 4, UNAM is far away from other institutions and it causes that all of them are seen into a single group; however, by eliminating UNAM, as it can be seen in the Figure 5,   In Figure 3 is shown that only one component represents around 81% of data variance of the research database. After PCA analysis and considering results in Figures 2 and 3, the graph that can be seen in Figure 4 was created; it might seem remarkable that the sum of the variation in this graph exceeds 100%. This is because, as commented in previous paragraphs, the CSMU database was separated into two databases corresponding to higher learning and research and then, reducing all the higher learning database dimensions to a single principal component that is projected on the ordered axis and reducing all the research database dimensions to a principal component which is projected on the abscissa axis; the graph was created maintaining a total data variance of 85% and 81% for every database respectively, for this reason the sum of both axes exceeds 100%.  In Figure 3 is shown that only one component represents around 81% of data variance of the research database. After PCA analysis and considering results in Figures 2 and 3, the graph that can be seen in Figure 4 was created; it might seem remarkable that the sum of the variation in this graph exceeds 100%. This is because, as commented in previous paragraphs, the CSMU database was separated into two databases corresponding to higher learning and research and then, reducing all the higher learning database dimensions to a single principal component that is projected on the ordered axis and reducing all the research database dimensions to a principal component which is projected on the abscissa axis; the graph was created maintaining a total data variance of 85% and 81% for every database respectively, for this reason the sum of both axes exceeds 100%. As it can be seen in Figure 4, UNAM is far away from other institutions and it causes that all of them are seen into a single group; however, by eliminating UNAM, as it can be seen in the Figure 5,   As it can be seen in Figure 4, UNAM is far away from other institutions and it causes that all of them are seen into a single group; however, by eliminating UNAM, as it can be seen in the Figure 5, a separation among the institutions becomes clear. At first glance the IPN, UAM and CINVESTAV appear to be the best institutions in research, whereas IPN and UdeG are the best universities in higher learning. It should be mentioned that the previous graphs are only representative of the total data in a certain percentage of the total available information, because, in the case of the component under research, PCA maintains a total data variance of 81% for research database, while in higher learning component, a total data variance of 85% for higher learning database is maintained.
Mathematics 2021, 9, x FOR PEER REVIEW 11 of 25 and UdeG are the best universities in higher learning. It should be mentioned that the previous graphs are only representative of the total data in a certain percentage of the total available information, because, in the case of the component under research, PCA maintains a total data variance of 81% for research database, while in higher learning component, a total data variance of 85% for higher learning database is maintained. After setting aside UNAM from the instance, the number of groups to cluster were determined using the silhouette coefficient method. It was applied over research and higher learning instances to determine the number of clusters. The comparisons for the three clustering techniques are on Figures 6 and 7. After setting aside UNAM from the instance, the number of groups to cluster were determined using the silhouette coefficient method. It was applied over research and higher learning instances to determine the number of clusters. The comparisons for the three clustering techniques are on Figures 6 and 7. Mathematics 2021, 9, x FOR PEER REVIEW 11 o and UdeG are the best universities in higher learning. It should be mentioned that previous graphs are only representative of the total data in a certain percentage of the t available information, because, in the case of the component under research, PCA ma tains a total data variance of 81% for research database, while in higher learning com nent, a total data variance of 85% for higher learning database is maintained.  The results given by the silhouette coefficient method show in Figures 6 and 7 t best number to classify the instances is 2. Moreover, this analysis is helpful for evaluat clustering performance where the highest values mean a better performance.

K-Means Results
This section may be divided by subheadings. It should provide a concise and pre description of the experimental results, their interpretation, as well as the experimen conclusions that can be drawn.
The result of this is that each cluster is associated not with a hard-edged sphere, with a smooth Gaussian model. Although GMM is categorized as a clustering algorith it is technically a generative probabilistic model describing the data distribution; due this property, there are two important limitations with GMM: the first one is about computation complexity because it is necessary to calculate the distributions, a whereby the algorithm can fail if the dimensionality of the problem is too high; the seco limitation is that in many instances, the number of groups is unknown and it may be n essary to experiment with a number of different groups in order to find the most suita Table 2 shows the classification with k-means. Items not included belong to quadr 1, which includes institutions that are not good in either of the two areas. It can be see first glance that there are institutions that remain there throughout time such as The N tional Polytechnic Institute (IPN), the Metropolitan Autonomous University (UAM) a the University of Guadalajara (UdeG); Likewise, in 2011 the Autonomous University Nuevo León (UANL) entered this zone and, like the previous institutions, remained th until the last year analyzed. In addition to the UANL, there are two more institutions t follow the same trend, the Monterrey Institute of Technology and Higher Educat (ITESM), which has been in this position since 2014, while the Meritorious Autonom University of Puebla (BUAP) does so from 2015. During the last year of analysis, the A tonomous University of Mexico State (UAMEX) was added to the list. Other instituti can be seen that also vary their position during the analyzed period; these transitions be observed in Figure 8, where the effort of the institutions to maintain or improve th status is evident throughout the study. The results given by the silhouette coefficient method show in Figures 6 and 7 that best number to classify the instances is 2. Moreover, this analysis is helpful for evaluating clustering performance where the highest values mean a better performance.

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.     Table 3 shows the classification with GMM and the elements not included belong to the static institutions, that is, minor in higher learning and research. It is evident that there are institutions that are maintained throughout the time such as: National Polytechnic Institute (IPN), Autonomous Metropolitan University (UAM) in the case of institutions of excellence; the UPN and UVM for the consolidated institutions in higher learning and the CINVESTAV as the only consolidated institution in research. Likewise, other institutions that are in transition can be observed, as shown in Figure 9, where the efforts of the institutions to maintain or improve their status is evident. It should be mentioned that the results of this classifier algorithm are quite similar to those of the k-means algorithm; however, perhaps the biggest difference is that the GMM appears to be more sensitive to increases and decreases in the databases.

• Excellence
Two of the invariable universities consolidated in excellence by the GMM method coincide with those obtained by k-means. For this method, there are two invariable universities consolidated in excellence, these are: IPN and UAM. •

Consolidated in higher learning
The results of the invariable universities consolidated in higher learning coincide with the results obtained by k-means since they only have UVM and NUP.

Consolidated in research
As with the k-means, the only institution regarded as invariable consolidated research is the CINVESTAV. It should be mentioned that the results of this classifier algorithm are quite similar to those of the k-means algorithm; however, perhaps the biggest difference is that the GMM appears to be more sensitive to increases and decreases in the databases.

• Excellence
Two of the invariable universities consolidated in excellence by the GMM method coincide with those obtained by k-means. For this method, there are two invariable universities consolidated in excellence, these are: IPN and UAM. •

Consolidated in higher learning
The results of the invariable universities consolidated in higher learning coincide with the results obtained by k -means since they only have UVM and NUP. •

Consolidated in research
As with the k -means, the only institution regarded as invariable consolidated research is the CINVESTAV.
• Static Using GMM, the list of institutions is rather similar to the one provided by k -means; the only changes are UAMOR, UGTO, UP and UNITEC. The first two leaves the first

Universities in Transition in GMM Analysis
• Universities that have improved in research.
In this analysis, the IBERO and the UASLP begin being static and during 2017 they were consolidated in research.
• Universities that became of excellence: Even though UdeG begins as part of the universities of excellence, it does not remain unchanged in its position, because during the period from 2013 to 2015, it is located with the consolidated institutions in higher learning, and it is until 2016 when it finally returns to be part of the excellence institutions. The ITESM begins as a consolidated university in higher learning, but its interest focuses on being part of the group of excellence; thus, by 2016 it achieves its goal and becomes a fourth quadrant institution. The UANL is a university with a behavior that shows its interest in being part of the g excellence group because it begins being consolidated in higher learning, for 2010 it becomes of excellence and although for the following year until 2015 it returns to the group where it started, in 2016 it is once again part of the group of excellence. The BUAP and UAMEX are institutions that, despite starting out as static, focused first on consolidating themselves in higher learning, a group to which they belonged from 2013 to 2015, and then gave the highest to the universities of the fourth quadrant in 2016.
• Universities that became static: The UABC, LASALLE and the UAS are institutions that begin as part of the static group, but there is an interest in consolidating themselves in higher learning, and although they achieve their goal in the period from 2013 to 2016; finally, in the last year of analysis, these institutions became static again. The UNITEC and UP are in a similar case, with the only difference that the period in which they are consolidated into higher learning corresponds only

Spectral Clustering Results
Regarding to the results obtained by the k-means and GMM algorithms, these have a certain relationship, since it turns out that both are consistent with each other; however, the results of the spectral grouping (see Table 4 and Figure 10) are complementary to the two techniques already mentioned and analyzed, this is because the spectral grouping emphasizes the changes in the less favored universities; where it is shown that some universities in the first quadrant want to improve, either in higher learning or in research, although their efforts are more modest.   A particularity of this area occurs in 2016, when the group with minor higher learning and major research had no members.

Conclusions
The analysis of the data with k-means shows congruent 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 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 congruent 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.

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