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Resumen Para lograr una evaluación más precisa =
de la
calidad del aire, es necesario conocer la relación que existe
entre las variables meteorológicas=
y los distintos contaminantes atmosféricos; =
lo que también tendrá la
finalidad de evitar los riesgos presentes tanto en el ecosistema como en la
salud de los humanos en un futuro próximo. El problema radica en
encontrar una asociación entre =
los contaminantes atmosféricos <=
/span>y las variables meteorológicas, que existe en los modelos y métodos d=
e categorización que pueden ser empleados. Debido a esta razón, el objetivo de este
artículo es analizar
la calidad de las asociaciones de contaminante=
s y variables meteorológicas por estaciona=
lidad
utilizando árboles de decisión para encontrar conocimiento que
permita localizar patrones que serán importantes para el análisis ambiental. Por consiguiente, al aplicar
árboles de decisión se pudo lograr un control periódic=
o de
la calidad de las asociaciones de contaminantes y variables meteorológicas, cuya validación del nivel de confianza de las reglas de asociación es mayor=
al
70% en los meses estudiados.
Palabras clave: Árboles de decisión, discretización, =
inteligencia artificial, minería d=
e datos,
reglas de
asociación.
Abstract To =
achieve a more accurate <=
span
class=3DSpellE>assessment of air quality , it is necessary to know
the relationship between meteorological va=
riables
and the different a=
ir pollutants ; this will also be aimed
at avoiding the risks present =
both in<=
span
style=3D'letter-spacing:-.65pt'> the ecosystem and in human
health in the future.
The problem
begins with finding an association
between air pollutants and meteorological variables in the<=
/span> models and categorization =
methods that can be used .=
Due to this
reason , the objective of this
article is to analyze the quality
of the associations
between pollutants =
and meteorological variables by seas=
onality
using decision trees to find
knowledge that allows locating patterns that will be important f=
or environmental analysis Therefore , it was possible to achieve a periodic c=
ontrol
of the quality
of the associations
of pollutants and <=
span
class=3DSpellE>meteorological variables, whose<=
/span> validation of the
confidence level of the association
rules is greater than 70%
in the months studied .
Keywords: Decision trees , discretization , artificial intelligence , data mining , <=
span
class=3DSpellE>association rule=
s.
Introducción Actualmente, la contaminació=
;n
atmosférica es uno de los problemas ambientales más graves del mundo, debido a la gran cantidad
de contaminantes emitidos
por los humanos, entre e=
llos destaca
el material particulado, menor a 2.5 micrómetros (PM2.5 ); el material particulado, menor a 10 micrómetros (PM10 ); el dióxido de azufre (SO2 ); el monó=
xido de carbono (CO);
el ozono (O3 =
); y el dióxido de nitró=
;geno (NO2 ). Estos contamin=
antes presentan un efecto dañino a la salud debido a que provocan diferentes afecciones pulmonares, cerebrales y cardiovasculares,
lo que reduce la esperanza de vida (Manisalidis=
et al., 2020). En las zonas rurales persiste un menor índice de contaminación en comparación con las zonas urbanas, debido a la alta densidad
demográfica, con=
juntamente con el desarrollo de la industria y el transporte. Las consecuenc=
ias
que conlleva están rel=
acionadas con problemas de salud, que traen consigo
preocupación en los gobiernos. Así, los=
gobernantes tienen que enfocarse en una adecuada
gestión de la calidad del aire, utilizando diferentes
estrategias y herramientas que permitan una correcta interpretación =
de
los datos obtenidos en la atmósfera.
Para el análisis de este ti=
po de
estudios es necesaria la recolección de los datos provenientes de las estaciones de monitoreo meteorológico continuo, con la finalidad de conocer
la calidad del aire y a su vez encontrar los problemas de
contaminación atmosférica en una zona. Además, =
la relación de los contaminantes atmosféricos <=
/span>con las variables meteorológicas, como la hume=
dad,
temperatura, velocidad del aire, entre otros, tienden a afectar el clima del
entorno, y dan como resultado el deterioro de la salud
de las personas (Whit=
eman et al., 2014). Del mismo modo, Ahmed et al. (2020) sostienen
que las concentraciones =
de contaminación se enfocan en zonas específic=
as de
interés o parques industriales. Huang et al. (2018) determinó
que, si no se toman las debidas precauciones, como el cambio de combustible=
s a
más limpios o la depuración de buses anticuados, la
contaminación atmosférica continuará en aumento de man=
era
considerable.
Siendo así, los gobiernos l=
ocales
se encuentran recolectando de manera constante los datos de las variables
meteorológicas y contaminantes atmosféricos, a través =
de
estaciones meteorológicas equipadas con sensores, que recopilan toda la información del ambiente para
&nbs=
p;
luego ser procesada y brindar conocimiento. Se utiliz=
an
técnicas de minería de datos para el tratamiento de la
información obtenida y el descubrimiento de patrones de comportamien=
to
como árboles de decisión, regresiones logísticas,
algoritmos genéticos, entre otros.
En este estudio, para realizar el
análisis de los patrones de comportamiento y crear modelos
de predicción con los datos de los contaminantes atmosféricos y variables meteorológicas
en la ciudad de Cuenca
– Ecuador, se utilizan los árboles de decisión con la finalidad de validar el =
grado
de confianza de las reglas de asociación obtenidas en un previo estu=
dio
(Orellana et al., 2021).
Este
artículo est&aac=
ute; estructurado de la siguiente manera: La Sección 2 presenta los trabajos
relacionados con métodos semejantes; la Sección 3 expone la metodología utilizada para llevar a cabo esta investigación; la Sección 4 explica los resultados que se han obtenido tras la aplicación
de la metodología; y la Sección 5 presenta las
conclusiones obtenidas y los trabajos futuros.
Trabajos Relacionados=
Exi=
sten investigaciones que estudian a las variables meteorológicas
y a los contaminantes atmosféricos
y permiten obtener información para distintos propósitos, como
predecir nuevo conocimiento para determinar el nivel de calidad de la
atmósfera y su posible gestión. A continuación, se exp=
onen
las investigaciones que están vinculadas a técnicas de miner&=
iacute;a
de datos que extraen los conocimientos más importantes en este campo=
.
En el estudio
realizado por García et al. (2020)
se utilizan métodos de miner&iacut=
e;a de datos
para cuantificar el impacto ambiental producido por una empresa Courier en =
Lima
y Callao, en Perú. Como punto inicial,
los autores aplican
métodos de clasificación a las variables y clu=
sterización
(OneRule y K-means )=
, lo
cual permite detectar los de mayor demanda de despachos (4 grupos) y determ=
inar
el nivel de emisiones de KgCO2 diarias, separadas por una
temporalidad mensual y por tipo de vehículo utilizado. En cuanto a la
investigación expuesta por Gayathri et al. (2020), propone
la utilización de un modelo
basado en técnicas de miner&iacut=
e;a de datos para la predicción de la contaminación del aire, específicamente la aplicación de algoritmo
de árbol de decisión C4.5 . Los autores descomponen dicho modelo en cinco etapas:
recolección de d=
atos, preprocesamiento de los datos,
árbol de decisión, datos de prueba,
predicción, con<=
span
style=3D'letter-spacing:-.15pt'> lo cual concluyen que el sistema
propuesto ayudará a mejorar la predicción.
Acorde a Siwe=
k
et al. (2016), se realizan los modelos de predicción a diferentes
contaminantes como PM10 , PM2.5 , NO2 , SO2 y O3. Sin embargo, uno de los contaminantes más nocivos para la salud hum=
ana es
PM2.5, debido a la acumulación en el sistema respiratorio, disminuyendo el correcto funcionamiento pulmonar y aumentando =
las enfermedades respiratorias. El método que utilizan para predecir el valor de PM2.5 es el de bosques aleatorios y los resultados demuest=
ran
que el modelo presentado es viable para aplicarlo en cualquier ciudad contaminada. Es importante<=
span
style=3D'letter-spacing:-.3pt'> considerar el índi=
ce de calidad del aire (AQI)
utilizado por los gobier=
nos
para determinar el nivel de ca=
lidad que se presenta en la atmósf=
era relacionados con los contaminantes. Asimismo, el trabajo realizado por Huang et al. (2018) utiliza algoritmos
de redes neuronales, basados en el índice PM2.5 , para llegar a pronosticar la contaminación del ambiente. La conclusi&oa=
cute;n obtenida
en este estudio
es que la únic=
a variable
atmosférica vinc=
ulada
con el contaminante PM<=
sub>2.5 es la humedad,
debido a que ayuda a la disipación del contaminante.
Otro
contaminante altamente perjudicial es el PM10 , de acuerdo con Althuwaynee et al. (2020), debido a que analizan l=
as
correlaciones que existen entre PM10 y otros contaminantes
&nbs=
p;
como el O3 y SO2 . Los datos son
obtenidos en Kuala Lumpur, Malasia, y en su análisis se observa que existe una relación lineal
directa y positiva
entre PM10 y SO2 , mientras
que existe una
relación semi lineal entre PM10 y O3 .
Continuando con las investigaciones
relacionadas con redes neuronales, está la realizada por Athira =
et al.
(2018), que utilizó el conjunto de datos de AirNet , de pronóstico de
contaminación del aire de cinco días del Centro Nacional de
Monitoreo Ambiental de China (C=
NEMC)
y datos meteorológicos del sistema global de predicción (GFS),
donde se aplican modelos de aprendizaje profundo, como=
red neuronal recurrente (RNN), redes de gran memoria a corto plazo (LSTM)
y unidades recurrentes cerradas (GRU)=
. En otra investigación, realizada por Birant (2011), se presentan los resultados de comparación de distintos algoritmos de árboles
de decisión como C4.5 ,
CART , NBTree BFTree LADTree ,<=
span
style=3D'letter-spacing:-.25pt'> REPTree &a=
acute;rbol aleatorio, bosques aleatorios, y árbol modelo log&iacut=
e;stico (LMT), para clasificar y predecir los niveles de emisión del SOx , utilizando los d=
atos
recopilados de alrededor de 800 instalaciones industriales en Izmir,
Turquía. Los autores
compararon individualmen=
te cada nivel de emisión del SOx (bajo, medio,
alto y muy alto), obteniendo los mejores resultados en la categoría =
de
bajo nivel. Así, se pudo
establecer que los niveles de emisión pueden clasificarse y predecirse con éxito en el 82.4% de
los casos.
Por=
otro <=
span
lang=3DES style=3D'letter-spacing:-.2pt'>lado, Martínez-España et al. (2018), analizaron diferentes técnicas de aprendizaje automático
para predecir los niveles del O3 en la región de Murcia,
España. Las técnicas analizadas son de bosques aleatorios,
árboles de decisión y K vecinos más próximos (KNN ); la técnica d=
e bosques
aleatorios fue la que más se ajustó=
. En los resultados obtenidos se puede señalar que entre los
parámetros que más influyen en la predicción del ozono=
se
encuentran variables climáticas relacionadas con la temperatura, la
humedad y el viento.
Así pues, en este estudio se
realiza un análisis de los patrones de comportamiento y la
creación de modelos
de predicción con los datos obtenidos de los contaminantes atmosféricos y
variables meteorológicas=
en la ciudad de Cuenca - Ecuador,
a través de árboles de decisión para ratificar el nivel de confianza=
de
las reglas de asociación previamente establecidas en trabajos previos.
Metodología En la presente investigació=
n, la
metodología que se utilizó para y se encuentra representada e=
n la Figura 1. Este modelo
es considerado uno de los más utilizados y sus cin=
co
fases comprenden un camino o proceso de aprendizaje.
A continuación, se presenta=
n, con
más detalle, las fases de la metodología ADDIE aplicadas a es=
ta
investigación:
•
Análisis: La propuesta de
esta investigación es realizar un análisis de la calidad de l=
as
asociaciones de contaminantes y variables meteorológicas por
estacionalidad, aplicando &aacu=
te;rboles de decisión a fin de verificar si las reglas de asociación expuestas en la investigación de Orellana et al. (202=
1)
se cumplen con su calidad.
•
Diseño: Considerando el anális=
is de la calidad
de las asociaciones de<=
span
style=3D'letter-spacing:-.6pt'> contaminantes y variables meteo=
rológicas, el principal objetivo
de este análisis es el cumplimiento de la calidad de las reglas
de asociación y así definir
el comportamiento de las reglas de asociación
correspondientes a los meses específicos. Es por esa razón qu=
e,
con
&nbs=
p;
este objetivo en mente, se ha definido trabajar con la herramienta de minería de datos aplicando árboles de decisión, que es una técnica de inteligencia artificial utilizando el s=
oftware RapidMiner Esta herramienta reduce el uso del código para el modelado de los datos, agilizando
el análisis <=
span
style=3D'letter-spacing:-.6pt'> y el procesamiento de grandes cantidades de datos.
•=
; =
Desarrollo: Para la selección de los parámetros necesarios, se=
procede
en función de
<=
span
lang=3DES style=3D'letter-spacing:-.1pt'>la recopilaci&o=
acute;n de los datos <=
span
lang=3DES style=3D'letter-spacing:-.4pt'> para definir un algoritmo con la mayor <=
span
lang=3DES style=3D'letter-spacing:-.1pt'>precisión posible.
• =
Implementación: Para
realizar la implementación, se obtuvo un conjunto de datos de los
contaminantes atmosféricos tomados entre el mes de enero a diciembre=
del
año 2018 en la ciudad de Cuenca, Ecuador. Para realizar el preproces=
amiento
de los datos, donde se realiza la depuración y limpieza, se
utilizó una plataforma de ciencia de datos denominada RapidMiner ; luego
se procedió a realizar las pruebas correspondientes.
• =
Evaluaci&o=
acute;n: Mediante el <=
span
lang=3DES style=3D'font-size:12.0pt;mso-bidi-font-size:11.0pt;line-height:1=
03%;
letter-spacing:-.25pt'> ingreso de <=
span
lang=3DES style=3D'font-size:12.0pt;mso-bidi-font-size:11.0pt;line-height:1=
03%;
letter-spacing:-.3pt'> los datos al <=
/span>software <=
span
lang=3DES style=3D'font-size:12.0pt;mso-bidi-font-size:11.0pt;line-height:1=
03%;
letter-spacing:-.2pt'>mencionado <=
span
lang=3DES style=3D'font-size:12.0pt;mso-bidi-font-size:11.0pt;line-height:1=
03%;
letter-spacing:-.2pt'>se <=
span
lang=3DES style=3D'font-size:12.0pt;mso-bidi-font-size:11.0pt;line-height:1=
03%;
letter-spacing:-.2pt'>determinaron los parámetros más
importantes del conjunto de datos, para generar un árbol de decisión correspondiente =
a cada mes y verificar que las reglas de asociación obtenidas
se cumplan o no en ciertos meses.
Figura 1 Diseño de la metodología ADDIE
Preproc=
esamiento de datos Como se conoce, al recolectar los =
datos
por medio de los sensores de las estaciones meteorológicas, estos pueden presentar anomalías o valores atípicos, ya sea por diversas razones
como interrupciones en la fuente energética o errores de
calibración en los sensores que están utilizando. Por lo tanto, la recolección de datos de los contaminantes atmosféricos es tomada cada diez minutos
y las variables mete=
orológicas con una frecuencia de cada minuto.
La recolección d=
e los
datos entre los meses de enero a diciembre del 2018 fue de 52,559 registros=
.
Est=
e conjunto de datos =
contiene atributos que son dependientes de otros, en otras =
palabras, los valores finales
de una variable depen=
den de la mezcla
de cualidades que sean partícipes en una determinada dimensión del estudio.
Cabe recalcar que se implement&oac=
ute;
la depuración de valores no encontrados, con la finalidad de descart=
ar
los datos que contienen valores vacíos o nulos. Luego de realizar el
preprocesamiento de los datos, se obtuvieron un total de 39,507 registros y=
se
eliminaron un total de 8,497 datos, para obtener valores consistentes de
comportamiento.
Aplicaci&o=
acute;n de técnicas de discretización La discretización es el procedimiento en el cual se divide el rango del atributo
continuo en intervalos. Cada uno de los intervalos se etiqueta como =
un
valor discreto, para luego a los datos originales asignarlos a los valores
discretos (Hemada & Vi=
jaya
Lakshmi , 2013).
Este proceso tiene un enfoque impo=
rtante
en el preprocesamiento de datos para su utilización en técnic=
as
de inteligencia artificial, ya que con la aplicación de un mé=
todo
eficaz de discretización no solo se puede reducir la demanda de memo=
ria
del sistema y mejorar la eficiencia de la minería de datos, sino que
también hace que el conocimiento extraído del conjunto de dat=
os
sea más sólido y fácil de entender (Hemada
& Vijaya Lakshmi,
2013).
Par=
a este estudio, es necesario aplicar la discretización con cantidades similares de datos =
en cada grupo para utilizar
las reglas de asociación presentadas por Orellana et al. (2021) y así analizarlas mediante
árboles de decisión para obtener sus respectivas medidas
de rendimiento.
Aplicac=
ión de árboles de decisión El algoritmo de árboles de
decisión es un modelo lógico parecido a un árbol binar=
io
construido con base en un conjunto de datos de entrenamiento. Este ayuda a predecir el valor de una variable objetivo mediante el=
uso
de variables predictivas. Este es el método ideal para la
clasificación de información y su posterior evaluación en sus diferentes escenarios (Alsagheer et al., 2017).
Resultados Al momento de tener un conjunto de datos amplio,
lo óptimo y recomendable es agrupar
los
datos que representan una misma unidad. Por lo tanto, se debe tomar en cuenta que la técnica
sea fiable por el hecho de obtener patrones para observar con m&aacu=
te;s
detalle el comportamiento de los datos.
T&eacut=
e;cnica
de discretización =
Para tener un mejor enfoque en la =
investigación,
se decidió crear varios conjuntos con una cantidad de datos
equilibrada, como se puede observar en la Figura 2. A través
de esta técnica de asociación, presentada por Orellana=
et
al. (2021), se crearon ocho conjuntos con distintos rangos de datos para
finalmente verificar las reglas de asociación.
Figura 2 Discretización por frecuencia
Reglas<=
span
style=3D'letter-spacing:-.2pt'> de
asociación Para este estudio se contó =
con
tres reglas de asociación previamente definidas por Orellana et al.
(2021) y mediante la discretización se obtuvieron las mismas reglas =
con
el conjunto de datos.
&nbs=
p;
La primera regla de asociación establece que al tener una temperatura que se encuentra en un rango superior a 18,=
950
℃, junto a un valor de la velocidad del viento superior a 3,150 m/h,
produce como resultado una humedad relativa baja; en otras palabras, los da=
tos
de la humedad relativa se ubican menor
igual a 45,5%.
En la Figura 3, se puede observar
la primera regla de
asociación.
Figura 3 Primera regla de asociación
Por otro lado,
la segunda regla
de asociación propone
que al presentarse un=
a medición baja de O3, menor a 5,538 μg/m3 y una humedad relativa alta, mayor a 88,5%, da como resultado
valores altos del punto de rocío. Esto se puede apreciar en la
Figura 4.
Figura 4 Segunda regla de asociación
Por=
último, la tercera regla =
de asociaci&oac=
ute;n plantea que, al presentarse <=
/span>una humedad relativa baja, menor a 45,5%, y una medida de=
l O3
alta, que supere el valor de 45 μg/m3 , el resultado que se
presenta es una temperatura alta, que puede sobrepasar los 18,950 ℃. =
como se puede observar en la Figura 5.
Figura 5 Tercera regla de asociación
Á=
;rboles de decisión La finalidad de aplicar los
árboles de decisión es validar el nivel de confianza de las
reglas de asociación presentadas en la subsección anterior. Por<=
span
style=3D'letter-spacing:-.45pt'> lo tanto, se realizó la validación
del nivel de confianza con árboles de decisión mensualmente,
desde enero hasta diciembre de 2018, y se observó la frecuencia con =
la
que se cumplen dichas reglas de asociación.
Se apli=
có
este método a la primera regla y se obtuvo lo siguiente:
• =
La <=
span
lang=3DES style=3D'font-size:12.0pt;mso-bidi-font-size:11.0pt;line-height:1=
03%;
letter-spacing:-.65pt'> primera regla no <=
span
lang=3DES style=3D'font-size:12.0pt;mso-bidi-font-size:11.0pt;line-height:1=
03%;
letter-spacing:-.65pt'> se <=
span
lang=3DES style=3D'font-size:12.0pt;mso-bidi-font-size:11.0pt;line-height:1=
03%;
letter-spacing:-.65pt'> cumpli&oac=
ute; en <=
span
lang=3DES style=3D'font-size:12.0pt;mso-bidi-font-size:11.0pt;line-height:1=
03%;
letter-spacing:-.65pt'> 3 de <=
span
lang=3DES style=3D'font-size:12.0pt;mso-bidi-font-size:11.0pt;line-height:1=
03%;
letter-spacing:-.65pt'> los =
12 <=
span
lang=3DES style=3D'font-size:12.0pt;mso-bidi-font-size:11.0pt;line-height:1=
03%;
letter-spacing:-.65pt'> meses analizados=
, por =
lo <=
span
lang=3DES style=3D'font-size:12.0pt;mso-bidi-font-size:11.0pt;line-height:1=
03%;
letter-spacing:-.65pt'> que =
representa=
el
25%, siendo los meses de junio, julio y agosto.
•
Existieron dos meses en los que=
se
pudo observar que la velocidad del viento no influyó para representar
los niveles bajos propuestos de humedad relativa como se puede observar en =
la
Figura 6.
Figura 6 Árbol de decisión aplicado en la primera regla
de asociación
En cuanto a la segunda regla de asociación, los resultados=
obtenidos fueron:
• =
Su medida de rendimiento fue del
66.7%; es decir, la regla llegó a cumplirse en ocho de los 12 meses. Sin embargo, cabe recalcar que en algunos
casos la medición obtenida de O3 es demasiado elevada,
por lo que no se puede llegar
a establecer con claridad lo que sucedió =
en el
mes de enero.
=
• =
También existe
el caso en el que los valores
de O3 llegan
a ser los esperados<=
span
style=3D'letter-spacing:-.35pt'> pero la
humedad relativa es baja y no llega a superar el 88.5%, lo cual influyó en la obtención
de la medida de rendimiento de la regla, como se puede observar en la
Figura 7.
Figura 7 Árbol de decisión aplicado en la segunda regla
de asociación
Por último, los resultados
obtenidos en la tercera regla de asociación, como se puede observar =
en
la Figura 8, fueron:
• =
En <=
span
lang=3DES style=3D'font-size:12.0pt;mso-bidi-font-size:11.0pt;line-height:1=
03%;
letter-spacing:-.75pt'> esta últ=
ima regla se <=
span
lang=3DES style=3D'font-size:12.0pt;mso-bidi-font-size:11.0pt;line-height:1=
03%;
letter-spacing:-.65pt'> trabaja con =
O3, humedad relativa y temperatur=
a, cumpli&eac=
ute;ndose
en el 75% de los meses excepto los meses de enero, julio y
octubre.
=
• =
Igual existen casos problemáticos con el contaminante O3 , d=
ebido a que los valores
medidos tienden a ser bajos, pero se alcanza la temperatura y humedad relat=
iva
planteada por la regla de asociación.
Figura 8 Árbol de decisión aplicado en la tercera regla de asociación <=
/i>
Conclusiones A lo largo de la investigaci&oacut=
e;n se
aplicaron técnicas de minería de datos,
siendo los árboles de decisión los que permitieron realizar un análisis del comportamien=
to de las variables meteorológicas y contaminantes atmosféricos, conjuntamente con las reglas
de asociación de=
forma
mensualizada. Así, los datos analizados fueron los recolectados por =
las
estaciones meteorológicas en la ciudad de Cuenca, Ecuador, del
año 2018. De esa manera, una de las etapas primordiales que se reali=
zaron
en esta investigación fue la discretización de los datos, deb=
ido a que fue necesario realizar
una correcta modulación de los datos
obtenidos para aplicar el algoritmo y obtener resu=
ltados
eficientes.
Par=
a concluir esta investigaci&=
oacute;n es importante tener en cuenta que los árboles de decisión son algoritmos de aprendizaje <=
/span>supervisado no paramétricos que permiten predecir el valor =
de una variabl=
e objetivo
mediante el uso de variables predictivas. Por lo tanto, al utilizar este
método en esta investigación hizo posible un control
periódico de la calidad de las asociaciones de contaminantes y variables meteorológicas por estacionalidad, siendo
así la comprobación de las
reglas de asociación en la mayoría de los meses
estudiados cumpli&eacut=
e;ndose más de 70% de ellas.
&nbs=
p;
Como un trabajo futuro, al tener un
conjunto de datos atmosféricos más amplio se puede llegar
a predecir mediante
la aplicación de árboles de decisión el comportamiento de los
contaminantes atmosféricos y presentar un tratamiento efectivo y
preventivo ante factores de riesgo que puedan afectar negativamente la salu=
d de
la población.
Reconocimientos Los autores desean agradecer al
Vicerrectorado de Investigaciones de la Universidad del Azuay por el apoyo financiero y académico, así
como a todo el personal de la escuela de Ingeniería de Ciencias de la
Computación, y el Laboratorio de Investigación y Desarrollo en
Informática (LIDI).
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