In this paper, the results of six years of research in engineering  education, in the application of the European Higher Education Area (EHEA) to  improve the performance of the students in the subject Analysis of Circuits of  Telecommunication Engineering, are analysed taking into consideration the fact  that there would be hidden variables that both separate students into subgroups  and show the connection among several basic subjects such as Analysis of  Circuits (AC) and Mathematics (Math). The discovery of these variables would  help us to explain the characteristics of the students through the teaching and  learning methodology, and would show that there are some characteristics that  instructors do not take into account but that are of paramount importanc

Bonache Perez, Jorge Luis

Hernandez Perdomo, Wilmar

Archivo Digital UPM

First steps in the discovery of patterns in the academic results of telecommunication engineering students in the subjects Analysis of Circuits and Mathematics

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Universidad Politécnica de Madrid (UPM)  Página 308 de 957 
 
 
 
Proceedings of Research in Engineering Education Symposium 2011 
Madrid, 4th - 7th October 2011 
 
First steps in the discovery of patterns in the academic results of telecommunication 
engineering students in the subjects Analysis of Circuits and Mathematics 
Wilmar Hernandez 
whernan@ics.upm.es 
Universidad Politécnica de Madrid, Madrid, 
Spain 
 
Jorge Bonache 
jbonache@euitt.upm.es 
Universidad Politécnica de Madrid, Madrid 
Spain 
Abstract: In this paper, the results of six years of research in engineering 
education, in the application of the European Higher Education Area (EHEA) to 
improve the performance of the students in the subject Analysis of Circuits of 
Telecommunication Engineering, are analysed taking into consideration the fact 
that there would be hidden variables that both separate students into subgroups 
and show the connection among several basic subjects such as Analysis of 
Circuits (AC) and Mathematics (Math). The discovery of these variables would 
help us to explain the characteristics of the students through the teaching and 
learning methodology, and would show that there are some characteristics that 
instructors do not take into account but that are of paramount importance. 
Introduction 
The last six years have experienced a change in the teaching and learning methodology at 
university, in which professor of basic subjects of Telecommunication engineering such as 
Analysis of Circuits (AC) have moved from a traditional way of teaching to the application 
of the European Higher Education Area (EHEA) in order to improve the performance of 
the students. 
The educational experiment presented in this paper was carried out in several stages. At 
the beginning, treatment and control groups were formed in AC, and some partial results 
were achieved. In the second stage, an analysis among all the subjects of first year students 
was carried out. The results of these two stages have been published in journals and 
international conferences on engineering education [Hernandez, Palmero et al. (2009); 
Hernandez, Palmero et al. (2010); Hernandez, Bonache et al. (2010)]. 
In [Hernandez, Bonache et al. (2010)], when conducting the statistical modelling of the 
student marks of AC, as a result, it was obtained that the observations were classified into 
two groups taking into consideration their probabilities of membership to these two 
groups . Hence, the threshold that divided both groups was found. Consequently, given the 
value x of X (mark), the threshold allowed us to decide which group an element belonged 
to. Now, we are interested to find the qualitative characteristics that distinguished the 
students of the groups that were obtained when classifying. 
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Proceedings of Research in Engineering Education Symposium 2011 
Madrid, 4th - 7th October 2011 
 
Taking into account the previous analysis for the marks, it can be said that there were 
some factors or hidden variables that were affecting significantly the academic results of 
the students and originating heterogeneity. That is to say, there were some factors that 
were producing a higher variability than the rest and, as a result, these factors were 
segmenting the population. 
Here, it is important to point out that for the case under analysis the only information 
available was the marks of AC, which was not enough to look for the qualitative factors 
that differentiated the students of the two groups. However, it was suspected that one 
important factor, among others, was the following: how Mathematical tools are used in AC. 
The academic results in Math should have an influence in the ones in AC and could be one 
of the reasons for segmenting the population. Hence, in order to continue improving the 
performance of the students in AC, it could be interesting to study whether the results in 
Mathematics I (MATI) have influenced the ones in Analysis of Circuits I (ACI), and if the 
marks in MATI represent a factor that segment the mark in ACI. 
Finite mixture of regression 
In the third stage, which is the current one, the marks and trajectories of the students that 
were majoring in Sound Systems (SM) and Telecommunication Systems (TM) in the 
subjects ACI and Math, have been collected (MATISM, ACISM, MATITM, ACITM) for six 
years and compared with each other. Instructors of both subjects (i.e., ACI and MATI) have 
worked together and new interdisciplinary materials of study have been created 
[Hernandez (2010)]. 
The study that has been carried out from the marks of the students has been the following: 
First, from the dispersion diagrams shown in Fig.1 and the calculation of the correlation 
coefficient (0.5345 for SM and 0.4822 for TM), it is observed both that there exists a linear 
relation between the independent variable MATISM (variable x) and the dependent 
variable ACISM (variable y), and that there also exists a linear relation between the 
independent variable MATITM (variable x) and the dependent variable ACITM (variable 
y). Hence, a linear regression model could be adjusted to the data. Nevertheless, when 
observing Fig. 1 deeply it can be seen that the marks in ACI are not homogeneous among 
students with similar marks in MATI. Therefore, it could be suggested that there are 
several groups for which a linear regression model would represent a good 
approximation. That is to say, there exists a different linear relation for groups of students 
for the level, the slope and the variability [Justel (2001)]. 
For the regression models it is assumed that the regression coefficients are the same for 
all the observations and it is also assumed that the sample (,)iiyx is a homogeneous group. 
In many cases, as it could be ours, the former assumption cannot be made if there are 
important variables that are not included in the model; that is, there is non-observed 
heterogeneity. 
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Proceedings of Research in Engineering Education Symposium 2011 
Madrid, 4th - 7th October 2011 
 
 
Figure 1: Dispersion diagrams: Marks of Math vs. ACI for students majoring in SM and TM 
Therefore, second, the model that is going to be specified is a finite mixture of regression 
models. That is, a set of K regression models [Justel (2001); Hurn et al. (2003); Frühwith-
Schnatter (2006); Bishop (2006); Leisch (2004)] 
 =  +  + ,    ~(0, 	
) 
characterized by their parameters: 
(, , 	
), … … . , (, , 	
) 
Thus, the density function of the mixture is given by 
(|, ) =  | + , 	



 
where  · | + , 	
 represents the density function of the normal with mean 
 +  and variance 	
, · | + , 	
 was called a component of the mixture o class, 
i stands for a priori 	

 stands for the set of all the 
parameters of the model. 
Third, all the pairs (, ) are observed and the following parameters are estimated:  
, … , , , … , , , … , , 	
, … , 	
, 0 <  < 1,   = 1


 
In order to estimate the model parameters, the first step is to estimate the number of K-
components of the model. To that end, the best model, Mi, will be chosen by using the 
Bayesian information criterion (BIC) [Hastie et al. (2001)]: 
 = () = 2 log () + () log  ,  = 1, … ,  
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Proceedings of Research in Engineering Education Symposium 2011 
Madrid, 4th - 7th October 2011 
 
where L(Mi) represents the likelihood function for parameters in Mi, evaluated at the 
maximum likelihood estimators, and () represents the number of parameters of the 
model . The model that will be chosen is the one with the smallest BIC. 
Here, all the calculations are carried out by using the flexmix Package of R [R-project for 
statistical computing; Leisch (2004); Grün & Leisch (2007)], and the following results are 
shown: 
 
After K is determined (K = 3), the second step is the estimation of the parameters. To that 
end, the EM algorithm is used [Bishop (2006); Hastie et al. (2001)] and a test for 
significance of regression coefficient [Grün & Leisch (2007)] is carried out. 
 
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Proceedings of Research in Engineering Education Symposium 2011 
Madrid, 4th - 7th October 2011 
 
 
Usual R convention: 
Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 · 0.1 1 
The above results, in both cases, tell us that know we should test a sub-model in which the 
ordinates in the origin of two components be equal to zero (for the case of SM) and the 
slope of the third component be equal to zero (for the case of TM). For SM and TM, the BIC 
of the sub-model is obtained and compared with the BIC that were previously obtained. 
 
Therefore, the sub-model for SM is chosen and the estimations of the parameters are the 
following: 
 
In addition, the sub-model for TM is chosen and the estimations of the parameters are the 
following: 
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Proceedings of Research in Engineering Education Symposium 2011 
Madrid, 4th - 7th October 2011 
 
 
The estimated a posteriori probability that the i-th observation belongs to the j-th 
component 1, 2, 3 is given by [Bishop (2006); Leisch (2004); Hastie et al. (2001)] 
 
The a posteriori probabilities can be used to build groups or clusters with the data 
assigning each observation to the component with maximum a posteriori probability. For 
the case under analysis, the marks are classified in three groups (clusters) shown in Fig. 2. 
 
Figure 2: Clusters that are built with the data assigning each observation to the component with 
maximum a posteriori probability 
 
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Proceedings of Research in Engineering Education Symposium 2011 
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and the percentages of the classifications of each group are the following: SM1: 66.32%, 
SM2: 23.12%, SM3: 10.54%, TM1: 65.14%, TM2: 22.02%, TM3: 11.36%. 
Conclusions 
To sum up, for SM and TM there exist three heterogeneous groups in the data and we 
would have the attachment of each student to each one of these groups, as well. Moreover, 
it has been observed that in TM there is a small group (cluster) of students whose marks 
tend to oscillate around a constant mark in AC (TM3). In addition, it has been observed 
that also there are two groups (clusters) consisting of most of the students (approximately 
2/3 parts) in which there would be a linear relation between the marks of AC and Math in 
SM and TM (SM3 and TM1). 
Furthermore, observing TM1 and TM2, it could be suggested that there exists a hidden 
variable that will allow us to explain the membership to each one of these groups, which 
would occur in a similar but not so clear manner when observing SM2 and SM3. For the 
case of SM1, it results that the students have approximately the same marks in Math and 
AC. Hence, it should have to be taken into consideration the fact that there would be 
hidden variables that separate students into subgroups. The discovery of these variables 
would help us to explain the characteristics of the students through the teaching and 
learning methodology, and would show that there are some characteristics that 
instructors do not take into account but that are of paramount importance. 
Future research plans 
In order to discover significant differences among engineering students, it is important to 
study both the information collected from the students and the questions that would be 
interesting to ask them, because there would be hidden factors that we want to discover. 
To be more specific, from the finite mixture model that has been adjusted to the data, in 
order to try to find the variables that explain the above-mentioned clusters, we think that 
there is some information about the students that have not been taken into consideration 
yet. For instance, it is important to know the number of times each student has taken the 
subjects AC and Math. Furthermore, as these subjects are first-year, first-semester 
subjects, it could be possible that subjects such as Physics and Mathematics taught 
previously to entering university are influencing the performance of the students in their 
first academic year at university. 
In addition, as during the educational experiment it was observed that there are groups 
whose marks oscillate around a constant value in the subject AC, this would suggest that 
these groups consist of students who have centered their preparation in Math and have 
probably abandoned AC. This abandonment could be analyzed by means of introducing 
the performance of the students during the course assignments of AC and Math, as another 
explanatory variable in the model. 
Finally, it is important to point out that this educational experiment is an ongoing work, in 
which the next stage would consist of trying to discover the existence of specific 
characteristics of each cluster. To sum up, we would try to discover a pattern in the data, 
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Proceedings of Research in Engineering Education Symposium 2011 
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which obviously would mean an improvement in the teaching and learning process in AC 
and Math. To this end, besides of the marks of the students in AC and Math, for the next 
stage of the educational experiment it would be necessary to manage to get additional 
information from the students. 
References 
Hernandez, W., Palmero, J., Labrador, M., Alvarez-Vellisco, A., & Bonache, J. (2009). 
Analysis of Results of Application of a student-center Learning System to Improve 
Performance of First-year Students, International Journal of Engineering Education, 
25(1), 161-172. 
Hernandez, W., Palmero, J., Labrador, M., Bonache, J., Cousido, C., Alvarez-Vellisco, A., 
Gutierrez-Arriola, J. M., & Jimenez-Trillo, J. (2010). Analysis of the results of four 
years of research and application of a studentcentered system based on the ECTS 
to first-year students in order to improve their performance in the subject AC-I, 
Proceedings of the 1st Annual Engineering Education Conference, IEEE EDUCON 
2010 (pp. 237-242). Madrid, Spain: IEEE 
Hernandez, W., Bonache, J., Cousido, C., Palmero, J., Labrador, M., & Alvarez-Vellisco, A. 
(2010). Statistical Analysis of Academic Results Before and After Four Years of 
Bologna. International Journal of Engineering Education, 26(6), 1493–1502. 
Hernandez, W. (2010). Herramientas Matemáticas para Análisis de Circuitos. Madrid, Spain: 
Dpto. Publicaciones de la EUIT de Telecomunicación. 
Justel, A. (2001). Estimación de mixturas de regresiones en el estudio de las emisiones de 
CO2 por países, Proceedings of Conferencia Internacional de Estadística en Estudios 
Medioambientales, EMA’01 (pp. 107-113). Cádiz, Spain: Universidad de Cádiz. 
Hurn, M., Justel, A., & Robert, C. P. (2003). Estimating Mixtures of Regressions. Journal of 
Computational and Graphics Statistics, 12(1), 55-79. 
Frühwith-Schnatter, S. (2006). Finite Mixture and Markov Switching Models. New York: 
Springer. 
Bishop, C. M. (2006). Pattern Recognition and Machine Learning. New York: Springer. 
Leisch, F. (2004). FlexMix: A general framework for finite mixture models and latent class 
regression in R. Journal of Statistical Software, 11(8), 1-18. 
Hastie, T., Tibshirani, R., & Friedman, J. (2001). The Element of Statistical Learning. New 
York: Springer. R-project for statistical computing. Accessed at www.r-project.org 
Grün, B., & Leisch, F. (2007). Fitting finite mixtures of generalized linear regressions in R. 
Computational Statistics & Data Analysis, 51(11), 5247-5252. 
  
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Acknowledgements 
This research has been supported by the Universidad Politécnica de Madrid under the 
research projects IE105902140, IE09590203, IE08590220 and IE075902067. 
Copyright statement 
Copyright © 2011 Wilmar Hernandez and Jorge Bonache: The authors assign to the REES 
organisers and educational non-profit institutions a non-exclusive licence to use this 
document for personal use and in courses of instruction provided that the article is used in 
full and this copyright statement is reproduced. The authors also grant a non-exclusive 
licence to REES to publish this document in full on the World Wide Web (prime sites and 
mirrors) on CD-ROM and in printed form within the REES 2011 conference proceedings. 
Any other usage is prohibited without the express permission of the authors. 
 
  


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First steps in the discovery of patterns in the academic results of telecommunication engineering students in the subjects Analysis of Circuits and Mathematics

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