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Entrenamiento de redes neuronales recurrentes usando la dimensión de encaje en el espacio de las fases

 

= Training of recurrent neural networks using the embedding dimension in the space the phases=

 


= 1

Nelly Patricia Peruga= chi Cahueñas

 

https://o= rcid.org/0000-0001-6331-9551

 

 

Escuela Superior Politécnica de Chimborazo (ESPOCH), Facultad de Administración de Empresa= s, Escuela de Transporte

patricia.perugachi@espoch.edu.ec

= 2

Mariela Micaela Moreno Palacios

 

https://o= rcid.org/0000-0002-1605-6442

 

 

Escuela Superior Politécnica de Chimborazo (ESPOCH), Facultad de Ciencias, GEAA=

= mariela.moreno@espoch.edu.ec

3

Arquímides Xavier Haro Velasteguí                       https://orcid.or= g/0000-0003-3391-5082

Escuela Superior Politécnica de Chimborazo (ESPOCH), Facultad de Ciencias, Escuela de Físi= ca

aharo@espoch.edu.ec

 

 

 

 

 

 

Artículo de Investigación Científica y Tecnológica

Enviado: 17/05/2022

Revisado: 28/06/2022

Aceptado: 01/07/2022

Publicado:29/07/2022

                             DOI: https://doi.org/10.33262/c= oncienciadigital.v5i3.1.2252

&nb= sp;

 

 

Cítese:

 

 

Perugachi Cahueñas,= N. P., Moreno Palacios, M. M., & Haro Velasteguí, A. X. (2022). Entrenamiento de redes neuronales recurrentes usando la dimensión de enca= je en el espacio de las fases. ConcienciaDigital, 5(3.1), 254-274. https://d= oi.org/10.33262/concienciadigital.v5i3.1.2252

 

 

CONCIENCIA DIGITAL, es una revista multidisciplinar, trimestral, que se publicará en soporte electrón= ico tiene como misión contribuir a la   formación de profesionales competentes con visión humanística y crítica que sean capac= es de exponer sus resultados investigativos y científicos en la misma medida= que se promueva mediante su intervención cambios positivos en la sociedad.&nb= sp;https= ://concienciadigital.org  

La revista es edita= da por la Editorial Ciencia Digital (Editorial de prestigio registrada en la Cámara Ecuatoriana de Libro con No de Afiliación 663) www.c= elibro.org.ec

 

= Esta revista está protegida bajo una licencia Creative Commons Attribution Non Commercial No Derivatives 4.0 Internatio= nal. Copia de la licencia: http://creativecommons.org/licenses/by-nc-nd/4.0/

Palabras claves: =

Teoría del Caos, redes neuronales recurrentes, dimensión de encaje, sistemas caóticos.

 

Resumen

Se propone un método usando Redes Neuronales Recurrentes para la predicción de datos caóticos, aplicando la Teoría del Caos, para estudiar el comportamiento dinámico de los datos en el espacio multidimensional de las fases, establecer la correlación de los mismos y determinar la dimensión de encaje como base para el entrenamiento de las redes neuronales, así como determinar las características dinámicas del sistema calculando los coeficientes de Lyapunov y la entropía de Kolmorov-Siani, que nos indican el grado de desorden que tiene el sistema, para proyectar la precisión de la predicción. Se usan datos de contaminan= tes PM2.5 tomados en el centro Histórico de la ciudad de Quito, en intervalos= de una hora, entre los años 2005 a 2019. Los resultados determinan que las series de datos corresponden a un sistema caótico (más de un coeficiente positivo de Lyapunov), por lo que se justifica la aplicación de la Teoría= del Caos en el análisis de los mismos, dando buenos resultados en las predicciones aplicando los métodos de redes neuronales recurrentes de Elm= an y Jordan, al comparar las series predichas se demuestran que no presentan diferencias significativas entre ellas, ni con los datos medidos, usando = el método de varianza con 0,05 de significancia,  el error cuadrático porcentual respecto al rango de variación de l= os datos es aproximadamente del 5 % en ambos casos. Objetivos: Proponer un método que ayude al entrenamiento de las redes neuronales usando la Teoría del c= aos, mediante la implementación de la dimensión de encaje en el espacio de las fases.

 

Keywords:

chaos theory, recurrent neural networks, embed= ding dimension, chaotic system.

 

Abstract

A method is proposed using Recurrent Neural Networks for the prediction of chaotic da= ta, applying Chaos Theory, to study the dynamic behavior of the data in the multidimensional space of the phases, establish the correlation of the sa= me and determine the embedding dimensions as a basis for the training of the neural networks, as well as determine the dynamic characteristics of the system by calculating the Lyapunov coefficients and the Kolmorov-Siani entropy,  that tell us the degree= of disorder that the system has, to project the accuracy of the prediction. = Data on PM2.5 pollutants taken in the Historic Center of the city of Quito, at one-hour intervals, between the years 2005 to 2019, are used. The results determine that the data series correspond to a chaotic system (more than = one positive Lyapunov coefficient), so the application of Chaos Theory in the analysis of them is justified, giving good results in the predictions applying the methods of recurrent neural networks of Elman and Jordan, wh= en comparing the predicted series they are shown that they do not present significant differences between them,   nor with the measured data, using the method of variance with 0.05 significance, the percentage square error with respect to the range of variation of the data is approximately 5% in both cases. Objectives: To propose a method that helps the training of neural networks using Chaos Theory, by implementing the socket dimension in the space of the phases.

 

 

 

Introducción

Uno de los problemas de las redes neuronales son l= os valores de entrenamiento de las series, lo0 cual se los puede tomar al azar, sin embargo, si se cambia el resultado puede ser diferente, por lo que en e= ste trabajo se propone juntar la Teoría del Caos y las redes neuronales para determinar ese valor, tomando datos de PM2.5 de la Red de monitoreo Atmosfé= rico de Quito (REMMAQ). En el año 2002 se implementó la Red Metropolitana de Monitoreo Atmosférico de Quito, para controlar los efectos de la contaminac= ión del aire (Rodríguez, 2015).

La teoría del caos es una rama de las ciencias que= se usa para estudiar sistemas complejos, sensibles a las condiciones iniciales= , en los cuales pequeñas variaciones en estas condiciones implican grandes diferencias en el comportamiento futuro, imposibilitando la predicción a la= rgo plazo. Esto sucede a pesar de que los sistemas son en rigor determinísticos= , es decir; su comportamiento puede ser completamente determinado conociendo sus condiciones iniciales (Fowler & McGuinness, 2010).

La red neuronal tiene un proceso concreto y tiende= a almacenar procedimientos prácticos y hacerlos valederos para su empleo (Reb= ala et al., 2019). Es comparable al cerebro del ser humano en dos perspectivas,= la primera, la red obtiene conocimientos mediante un procedimiento de preparac= ión; la segunda, los enlaces inter-neuronales llamadas cargas sinápticas muestran gran estabilidad en almacenar los conocimientos (López & Caicedo, 2009). Las redes neuronales recurrentes logran el comportamiento dinámico realimen= tando sus entradas con muestras anteriores de las salidas, el comportamiento diná= mico de las redes recurrentes hace que sean una poderosa herramienta para simula= r e identificar sistemas dinámicos no lineales o complejos (Díaz, 2016; Gonzále= z et al., 2017).

Con estos antecedentes y con el objetivo de establ= ecer una herramienta operacional para la gestión ambiental, se aplicarán estas teorías para analizar la variación y predecir las concentraciones de PM2.5 = en el Centro Histórico de la ciudad de Quito.

Metodología

Teoría del caos=

El desarrollo de la teoría del caos se dio hacia finales del siglo XIX, cuando el físico–matemático Henri Poincaré introdujo= el concepto de no linealidad, que contradecía el ideal clásico de los sistemas lineales, en los que causa y efecto se identifican plenamente y la resultan= te de un fenómeno es igual a la suma de sus partes. La no linealidad implica divergencia entre resultado y origen, y la simple adición de las partes de = un hecho no corresponde al producto final, por lo que los algoritmos lineales = poco pueden hacer para explicar su dinámica. El problema no era menor, si se considera que un 90% de los procesos de la naturaleza son de carácter no li= neal (Motter & Campbell, 2013).

Para la aplicación de la teoría del caos se sigue = un procedimiento que va desde la reconstrucción de las series de datos al espa= cio de las fases, la determinación del grado de casticidad que tiene el sistema= y llegando a la predicción de dichos sistemas, para lo cual se usa el software Tisian (Hegger et al., 1999).

Tiempo de retardo (delay time) y dimensión de encaje (ebedding dimention)

El tiempo de retardo, es el paso más importante en= el análisis del espacio de fases, es la técnica de reconstrucción del sistema, formando vectores y ejecutando una proyección en el espacio de fases, que e= s un espacio físico que permite detectar variables dinámicas, en este método se busca la relación biunívoca de los datos en el espacio de las fases (Ivance= vic & Ivancevic, 2007).

La técnica de reconstrucción de espacio de fase más importante es el método de retardo. Los vectores en un nuevo espacio, el espacio de empotramiento, se forman a partir de los valores retardados de l= as mediciones escalares expresado en la ec.1:

sn  =3D (sn -(m-1) <= span style=3D'font-size:12.0pt;line-height:200%;font-family:"Times New Roman",se= rif; mso-ansi-language:ES-EC;mso-bidi-font-weight:bold'>τ= , sn-(m-2) τ= , ... , sn ) =                                                 =                             (1)

El número m de elementos se denomina dimensión de encaje, el tiempo τ se denomina generalmente retardo. Los teoremas incorporados por Takens y Sauer (Abderrahim et al., 2014), afirman que, si = la serie sn consiste en mediciones escalares del estado de un sistema dinámico, entonces bajo ciertas suposiciones, la incorporación de tiempo de retardo proporciona una imagen uno a uno del conjunto original, siempre que m sea lo suficientemente grande.

Si N mediciones disponibles, el número de vectores= de encaje se estima que será N-(m-1) τ. Esto debe tenerse en cuenta para = la correcta normalización de las cantidades promediadas. Existe literatura sob= re la opción "óptima" de los parámetros de encaje m (dimensión de encaje) y τ (tiempo de retardo).

Fraser y Swinney el método de información mutua pa= ra calcular el retraso o llamado también tiempo de retardo con resultados razonables, tiene en cuenta las correlaciones no lineales, que se puede calcular con la Ec. 2 (Ivancevic & Ivancevic, 2007):<= /p>

                                                                =      (2)

Dónde para una partición en los números reales  es = la probabilidad de encontrar un valor de serie temporal en el intervalo i-ésim= o, y es la probabilidad conjunta de que una observación caiga en el intervalo i-ésimo y la posterior observación un tie= mpo <= span lang=3DES style=3D'font-size:11.0pt;line-height:115%;font-family:"Calibri",= sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:5.5pt;mso-text-raise:-5.5pt;mso-ansi-language:ES; mso-fareast-language:ES;mso-bidi-language:AR-SA'> , caiga en el j-ésimo intervalo. En teoría, = esta expresión no tiene dependencia sistemática del tamaño de los elementos de la partición y se puede calcular fácilmente. Existen buenos argumentos de que,= si la información mutua exhibe un mínimo marcado en un cierto valor, entonces = este es un buen candidato para un retraso razonable (Valentin, 2009).

La idea del algoritmo del falso vecino es que para cada punto  en = la serie de tiempo busque su vecino más cercano en un espacio m-dimensional. Por lo tanto, calcula la distancia , que reincide en ambos puntos. Como se expr= esa en la ec. 3.

                                                                    =                     (3)

Al exceder  un = umbral determinado , este punto está marcado como un falso veci= no. El criterio de que la dimensión de encaje es lo suficientemente alta, es la fracción de puntos para la cual >  es = cero, o al menos suficientemente pequeño. La introducción del concepto de falso vecino más cercano permite especificar una mínima separación temporal de los vecinos válidos y así determinar las dimensiones apropiadas para las matric= es creadas, buscando vecinos a un punto fijo y que estén contenidos dentro de = una bola de radio centrada en ese punto, de esta forma se determina la distancia interactiva y se calcula. Si la relación de la distancia de la interacción y del vecino más cercano excede a un valor determinado, el punto se marca com= o un vecino equivocado (Valentin, 2009).

Reducción del ruido

El estudio de series de datos puede llevar a sacar conclusiones erróneas ya que los datos que se toman como magnitud de la variable a analizar suelen estar influidas por varios factores que introduc= en distorsiones (ruido) y por lo tanto no permiten observar con claridad los patrones dinámicos que se quiere analizar. Por lo tanto, es necesaria la transformación de los datos de manera de poder separar el componente ruidos= o y extraer señales que informen acerca del comportamiento evolutivo de la serie (Fernández, 2014).

Dentro del análisis de series temporales, algunos métodos para la reducción del ruido usan el hecho que los puntos, es decir = los datos, están relacionados dinámicamente y que debido a hechos determinístic= os los puntos en una misma región del espacio tienen un comportamiento similar= . El algoritmo no lineal más simple sobre reducción de ruido consiste en reempla= zar la coordenada central de cada vector de encaje, por un promedio local de es= ta coordenada (Pino-Vallejo et al., 2018). Como se expresa en la Ec. (4):=

Al escoger una m dimensión de encaje y un adecuado tiempo de retardo τ, por cada vector de encaje Sn se forma una vecindad Une, se forma en el espacio de fase que contiene todos los puntos Sn tal que ║Sn – S´║˂ Ԑ, donde Ԑ es el radio de la vecind= ad, que debe ser lo suficientemente grande para cubrir la extensión de ruido y = S´ el espacio de fase inicial (Pino-Vallejo et al., 2018). Por cada vector de encaje Sn =3D (Sn-(m-1),....Sn) con λ=3D1, la coordenada corregida es Sn-(m/2), se calcula como el promedio sobre la vecindad Une.

Exponentes de Lyapunov=

El cálculo del espectro completo de Lyapunov requi= ere considerablemente más esfuerzo que el exponente máximo (Bochi & Rams, 2016). El proceso consiste en una estimación de los Jacobianos locales, es decir, de la dinámica linealizada que gobierna el crecimiento de las perturbaciones infinitesimales. El caos surge del crecimiento exponencial de las perturbaciones infinitesimales es así como para garantizar el análisis = de esta inestabilidad están los exponentes de Lyapunov, que cuantifican el mom= ento que dos trayectorias se alejan, como se expresa en la Ec 5.

                                                 =                                         (5)

Donde: r(t) es el semi-eje mayor de una elipse, r = es el radio pequeño y t es el tiempo grande, en general se establece λ conocido como coeficiente de Lyapunov. En un sistema caótico al menos un exponente de Lypunov es positivo.

El cálculo del espectro Lyapunov requiere considerablemente más esfuerzo que sólo el exponente máximo. Un método muy usado es la estimación de los jacobinos locales, es decir, de la dinámica linealizada, que gobierna el crecimiento de perturbaciones infinitesimales.= Se encuentra a partir de ajustes directos de modelos lineales locales del tipo= , de modo que la primera fila del jacobino e= s el vector , y  para i=3D2,…,m donde m  es la dimensión = de incrustación o encaje. El  se = halla con el método de los mínimos cuadrados  don= de  es = el conjunto de vecinos de  (Pino-Vallej= o et al., 2018).

La Entropía de Kolmogórov-Sinai, también llamada métrica (h), mide la pérdida de información a lo largo de la evolución del sistema. Como se expresa en la Ec. 6, este tipo de entropía es determinada = por la suma de exponentes positivos de Lyapunov. Tiene importante significado cuando se aplica a sistemas de los cuales no se dispone más que de series temporales de valores de cierta variable (Pino-Vallejo et al., 2018).

(6)

Este tipo de entropía se relaciona con la previsibilidad que expresa una escala de tiempo promedio hasta la cual las predicciones pueden ser posibles. Esta cantidad invariante intrínseca del sistema tiene una relevancia singular para la caracterización de la previsibilidad. Los valores más altos de h implican valores más bajos de la predictibilidad 1/h, es decir, cuanto más caótico, menos predecible (Domański & Ławryńczuk, 2017).

También de estos coeficientes se puede determinar = la dimensión fractal usando el método conocido de Kaplan – Yorke, 1979, Ec. 7 (Medvinsky et al., 2017; Stolz & Keller, 2017).

                                                       (7)

Redes Neuronales

Una red neuronal está construida por un conjunto de nodos interconectados que se denominan neuronas y se organizan en capas. Una red neuronal básica tiene una capa de entrada (input layer), una capa intermedia oculta (hidden layer) y una capa de salida (output layer). Los n= odos de la capa de entrada corresponden con el número de variables que queremos introducir en la red (serían los equivalentes a las variables independiente= s en una regresión lineal). Por otro lado, el número de nodos de salida depende = del número de variables a predecir. Por último, la capa intermedia se usa principalmente para aplicar transformaciones no lineales a las variables de entrada originales (Kocak et al., 2020).

 

 

 

 

Figura 1

Ejemplo de una red neuro= nal totalmente conectada

Fuente: Lewis (2016)

Las redes neuronales dado a su naturaleza y fundamentos poseen un gran número de rasgos similares a las neuronas del cerebro. Tienen la facultad de instruirse de la experiencia, como de sistematizar sucesos pasados y para poder llevarlos a nuevos sucesos (Cruz = et al., 2007).

Redes Neuronales Recurrentes=

Las redes neuronales recurrentes (Recurrent Neural Networks) tienen una gran habilidad para aprender problemas difíciles que tratan sobre datos con series temporales. Estas redes tienen estados ocultos distribuidos en el tiempo que permiten almacenar mucha información sobre el pasado de forma eficiente (Géron, 2017).

Las redes recurrentes son dinámicas, es decir tamb= ién tienen conexiones hacia atrás. La salida depende de los actuales valores de= las variables de entrada a la red, de los anteriores valores de la entrada, de = los anteriores valores de la salida, etc.

Fijémonos en la más sencilla red neuronal recurren= te posible, compuesta por sólo una neurona que recibe los inputs y que produce= un output y que este lo envía a sí misma como vemos en la parte izquierda de la siguiente imagen:

 

 

 

Figura 2<= /o:p>

Neurona recurr= ente (izquierda), desarrollo en el tiempo (derecha)

Fuente: Lewis (2016)

Cada neurona recurrente tiene dos conjuntos de pes= os; uno para los inputs x(t) y otro para las salidas del paso temporal anterior y(t-1). Llamemos a estos pesos wx y wy. La salida y(t) de una neurona recurrente se puede calcular de la siguiente manera, Ec. 9:

=                                                  =                    (9)

donde b es el sesgo (bias) y φ es la función = de activación. Podemos resumir la expresión anterior mediante el álgebra linea= l de la siguiente manera, Ec. 10:

                     (10)=

siendo W =3D [Wx Wy]T y donde:

Y(t) es una matriz m x n neuronas que contiene los outputs del marco temporal t para cada batch (que es el conjunto de datos q= ue entran a la neurona) en este caso m y n neuronas es el número de neuronas d= e la capa.

X(t) es una matriz m x ninputs que contiene las variables inputs de las m observaciones y ninputs es el número de variables explicativas de entrada.

Wx es una matriz ninputs x nneuronas conteniendo los pesos de las conexiones para las entradas del paso temporal actual.

Wy es una matriz nneuronas x nneuronas conteniendo los pesos de las conexiones para las salidas del paso temporal anterior.

Las matrices de pesos Wx y Wy se pueden juntar en la matriz W que será (n inputs +n neuronas) x n neuronas <= o:p>

b es un vector de tamaño neuronas que permite añad= ir un término constante (bias) al modelo.

Hay que darse cuenta de que Y(t) es una función de X(t) y de Y(t-1), quien a su vez es función de X(t-1) y de Y(t-2), quién es función de X(t-2) y así sucesivamente. Esto hace que Y(t) sea función de to= das las variables de entrada desde el tiempo inicial t =3D0 (esto es, de X (0);= X (1); : : : ;X(t) ). En t =3D0 no hay outputs anteriores luego se asumen como cer= os.

Redes Neuronales Recurrentes de E= lman y Jordan

Las Redes Neuronales Recurrentes son adecuadas para modelar series temporales, se trabajó con redes Elman y Jordan, el análisis= se realizó en el software libre R-Studio.

Una red neuronal de Elman es una red con tres capas (entrada, oculta y salida) a las que se le añade una capa con ’unidades de contexto’ (o también llamadas unidades de retardo). Cada capa contiene una o más neuronas que propagan la información de una capa a la siguiente con la regla de aprendizaje que se haya fijado.

En las redes de Elman el número de neuronas de la = capa de contexto tiene que ser igual al número de neuronas de la capa oculta. Además, todas las neuronas de la capa de contexto están conectadas con una neurona de la capa oculta.

El proceso de memoria se produce a través de las unidades de retardo que son alimentadas por las neuronas de la capa oculta.= Los pesos de las conexiones entre la capa oculta y las unidades de retardo son fijos e iguales a 1. Gracias a esto en las unidades de retardo siempre mantenemos una copia de los valores de salida de las neuronas de la capa oc= ulta del paso previo.

Podemos ver una representación de la red neuronal = de Elman en la figura 3:

 

 

 

 

 

 

 

 

 

 

Figura 3<= /o:p>

Representaciones de la red neuronal de Elman

Fuente: Lewis (2016)

En la figura 3 se ve el esquema de la red de Elman, observando que la salida de la capa oculta se almacena en la capa de retard= o, y que luego esta sirve de entrada en el siguiente paso para la capa oculta, d= onde las unidades de retardo ui en las que se almacena el valor de la salida de = la neurona yi y sirven como entrada en el siguiente paso (Lewis, 2016).

La red neuronal de Jordan tiene una arquitectura similar a la de Elman, es decir, consta de la capa de entrada, una capa ocu= lta, la capa de salida y la capa de contexto. La única diferencia es que ahora se guarda la información de la capa de salida en vez de la información de la c= apa oculta, por lo que la capa de contexto está conectada a la capa oculta y a = la de salida como vemos en la figura 4. Por tanto, esta red recuerda la salida final del paso anterior en vez de la salida de la capa oculta. La estructur= a de la red neuronal de Jordan es la siguiente:

Figura 4<= /o:p>

Representaciones de la red neuronal de Elman

Fuente: Lewis (2016)

Las ecuaciones matemáticas que definen ambos proce= sos son Ec. 11 y 12:

Red de Elman:

    =                                                 =                              (11)

Red Jordan:

                                                       =                         (12)

Donde xt es el vector con las variables= de entrada, ht es el vector de la capa oculta, yt es el vector de salida, W =3D [Wx Wy]T y Uh son las matric= es de pesos y b =3D [bh by]T el bias. σh y = σy son las funciones de activación de las respectivas capas.=

Para la presente técnica no es obligatorio describ= ir supuestos como la distribución de probabilidad, patrón de comportamiento de= los datos de la serie para pronosticar de manera eficiente, además al ser utili= zada para realizar predicciones presenta una característica significativa y es q= ue los datos no serán desarrollados para probar supuestos como tendencia o estacionalidad en la serie previo a la realización de los pronósticos (Li et al., 2017).

Es necesario realizar un cambio de escala a los da= tos originales debido a que los valores de la ecuación logística se encuentran = en el rango de (0,1), se aplica la siguiente formula:

                                                 =                                              (13)

Se ha estableció un ritmo de aprendizaje de 0.07 (= Zhou et al., 2017), con un número máximo de 5000 iteraciones. =

El uso de la Teoría del Caos ayuda a establecer el comportamiento dinámico del sistema y establecer su regularidad en el espac= io de las fases, además se reduce el ruido en la serie datos, los cuales se normaliza y se aplica redes neuronales recurrentes.

Resultados y discusión

Con los datos recolectados se procedió a desarrollar el análisis, en primera instancia una reducción de ruido en la serie de datos, con los que se traba= jó para establecer la dinámica del sistema, procediéndose antes a normalizar l= os datos, y finalmente se predijo con redes neuronales recurrentes, para lo cual se ha usado el modelo TISIAN y el paquete R estudio.

Figura 5

Gráfico de datos PM2.5 con reducción de ruido y sin reducción del ruido

En la figura 5 se observa los datos con reducción del ruido y sin reducción de ruido, donde no se puede notar diferencias significativas entre las series, procedimiento necesario para evitar anomalías entre los datos y asegurar su correlación en el espacio de las fases.

En principio para realizar el análisis se normalizó los datos entre cero y uno, para mantener la variación de estos según los requerimientos de las redes neuronales recurrentes, verificándose que aporta al análisis usando la teor= ía del caos, ya que los rangos de variación no son muy grandes, además, por la cantidad de datos usados el sistema se muestra estable en los parámetros calculados.

Figura 6<= /o:p>

Estimación del tiempo de retardo usando el método de información mutua

En la figura 6 se presentan los cálculos aplicando el método de información mu= tua para hallar el tiempo de retardo, para lo cual se recomienda elegir el prim= er mínimo de la función.

Figura 7<= /o:p>

Estimación de la dimensión de encaje usando el mét= odo de falsos vecinos

En la figura 7 se observa los resultados del cálculo de la dimensión de encaje usando el método de los falsos vecinos, el cual toma como valor más probabl= e el primer cero que alcanza la función.

Tabla 1

Tiempo de retardo y dimensión de encaje con reducc= ión del ruido

 =

PM 2.5

8

7

En la tabla 1 se observa el tiempo de retado para = la reconstrucción de los datos en el espacio de las fases y la dimensión de en= caje que establece la dimensión mínima que se debe usar para analizar los datos observados.

 

 

 

 

 

 

Figura 8<= /o:p>

Gráfico de datos los coeficientes de Lyapunov calculados

La figura 8 presenta los coeficientes de Lyapunov, indican la dinámica del sistema, cuando existe al menos un coeficiente positivo, se dice que el sistema es caótico, es decir son datos no lineales= , en los cuales es conveniente aplicar métodos propuestos, Teoría del Caos y Red= es Neuronales.

Tabla 2

Dimmensión fractal de Kaplan-Yorke y entropía de Kolmogoróv-Sinai

Parámetros<= /p>

Dimensión Fractal<= /p>

Entroía

PM2.5

         5,12

       0,0321

La tabla 2 muestra la dimensión de Kaplan-Yorke y = la Entropía de Kolmogórov-Sinai, del sistema que identifica la dimensión fract= al de este y el grado de desorden que los caracteriza.

Se determinó que la dimensión es fraccionaria, es decir es de tipo fractal, de 5,12 que ratifica la complejidad del sistema, de igual manera se aprecia la entropía del sistema que mide el grado de desorden del sistema y está asoci= ada a su predictibilidad.

 

 

Figura 9 

Gráfico de datosPM2.5 medidos y predichos con redes neuronales recurrentes de Elman

En la figura 9 se observa la comparación entre los datos reales y predichos us= ando redes neuronales recurrentes de Elman, tomando como base la dimensión de 7 determinada con la teoría del caos en la cual se describirían los datos con= un error de 0,07 y 5000 repeticiones en el entrenamiento de las redes, se puede notar una tendencia muy buena entre los datos reales y datos predichos.

Figura 10

Datos PM2.5 medidos y predichos con redes neuronales recurrentes de Jordan

En la figura 10 se observa la comparación entre los datos reales y predichos usando redes neuronales recurrentes de Jordan, tomando como base la dimensi= ón de 7 determinada con la teoría del caos en la cual se describen los datos c= on un error de 0,07 y 5000 repeticiones en el entrenamiento de las redes, simi= lar a lo aplicado en Elman, de igual forma se puede notar una tendencia muy bue= na entre los datos reales y datos predichos.

Tabla 3 =

Análisis de varianza de los datos predichos

Origen de las variaciones

Suma de cuadrados

Grados de libertad

Promedio de los cuadrados

F

Probabilidad

Valor crítico para F

Entre gru= pos

 

 

 

0,461

0,631

3,011

Dentro de= los grupos

 

 

 

Total

1,917

581

También se aplica un análisis de varianza en la ta= bla 3, para determinar si existe diferencia entre las series de datos tanto predic= has con redes neuronales de Elman, como de Jordan, obteniéndose que efectivamen= te no existe diferencia entre estas series.

Tabla 4

 Error cuadrático medio entre series de d= atos reales con predichos con la teoría del caos y Redes Neuronales Recurrentes<= o:p>

Método

Error cuadrático medio

Error cuadrático medio porcentual respecto al rango de variación.<= o:p>

Método de Elman

0,08062

4,98

Método de Jordan

0,08003

4,94

Finalmente se determinó el error cuadrático medio entre los datos predichos y los datos medidos (tabla 4), y se calculó el error porcentual que se daría en el rang= o de variación de la serie medida, para tener una idea de la precisión del cálcu= lo llegando en ambos casos alrededor de 4,9 %, demostrando que cualquiera de e= llos podría ser aplicado en este sistema.

Conclusiones

·&nb= sp;        Se aplica teoría del caos a las series de datos normalizados con reducción de ruido, obteniéndose resultados estables en los parámetros de análisis de la Teoría del Caos, con una dinámica caótica (2 coeficientes positivos de Lyapunov), que indica la irregularidad de los dat= os.

·&nb= sp;        Se establece la entropía de Kolmorov-Siani (0,0321= ) y la dimensión fractal de Kaplan-Yorke (5.2), resultados que confirman la complejidad del sistema y la dificultad en su predictibilidad.

·&nb= sp;        Al aplicar la dimensión de encaje como variable de entrenamiento de las redes se establece, que los dos métodos propuestos no presentar diferencias significativas al 0.05 de significancia, y su error porcentual medio cuadrático respecto al rango de variación de los datos de alrededor del 5 %.

·&nb= sp;        El conocimiento de la dinámica del sistema ayuda a preestablecer el comportamiento de los datos antes de predecir con redes neuronales recurrentes, dando información valiosa para saber las potenciales resultados.

 

Agradecimiento

Al Proyecto de Monitoreo Hídrico en la Provincia de Chimborazo del Centro de Energías Alternativas y Ambiente (CEAA).

 

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