Redes Neuronales - Introduciondonos en la Inteligencia...

Redes NeuronalesIntroduciondonos en la Inteligencia Artificial

Xavi Barber

Centro de Investigación OperativaUniversidad Miguel Hernández de Elche

Xavi Barber (@umh1480 @XaviBarberUMH) Redes Neuronales 1 / 67

1 El sistema nervioso

2 Redes Neuronales Artificiales

3 Métodos de Regresión y Redes Neuronales

4 Neural Networks y R

5 Variable importance


El sistema nervioso

El sistema nervioso


El sistema nervioso

Primeras Definiciones

Figure 1Xavi Barber (@umh1480 @XaviBarberUMH) Redes Neuronales 4 / 67

El sistema nervioso

Son un tipo de células del sistema nervioso cuya principal función es la excitabilidadeléctrica de su membrana plasmática. Están especializadas en la recepción deestímulos y conducción del impulso nervioso (en forma de potencial de acción)entre ellas o con otros tipos celulares como, por ejemplo, las fibras musculares de laplaca motora.


El sistema nervioso

Núcleo: Situado en el cuerpo celular, suele ocupar una posición central y es muyvisible, especialmente en las neuronas pequeñas. .

Dendritas: Las dendritas son ramificaciones que proceden del soma neuronal queconsisten en proyecciones citoplasmáticas envueltas por una membrana plasmáticasin envoltura de mielina.

Axón: Es una prolongación del soma neuronal recubierta por una o más células deSchwann en el sistema nervioso periférico de vertebrados, con producción o no demielina.


El sistema nervioso

Figure 2


Redes Neuronales Artificiales




Aplicaciones reales

Negocios y FinanzasLas redes neuronales se aplican en analisis financiero para la asignación derecursos y scheduling. Se aplican en Data Mining, esto es, recuperación dedatos y busqueda de patrones a partir de grandes bases de datos.

Evaluación de CréditosEjemplo: La empresa HNC de R. Hecht-Nielsen, ha desarrollado aplicaciones deredes neuronales para calcular el ranking de créditos (Credit Scoring). Elsistema HNC se aplica también para calcular el riesgo hipotecario. La empresaNestor aplica redes neuronales para aplicar seguros de manera automática en laconcesión de préstamos hipotecarios con un 84% de éxito aproximadamente.



Introducción

Las Redes Neuronales (ANN: Artificial Neural Networks) surgieronoriginalmente como una simulación abstracta de los sistemas nerviososbiológicos, constituidos por un conjunto de unidades llamadas neuronasconectadas unas con otras.El primer modelo de red neuronal fue propuesto en 1943 por McCulloch y Pittsen términos de un modelo computacional de actividad nerviosa. Este modeloera un modelo binario, donde cada neurona tenía un escalón o umbralprefijado, y sirvió de base para los modelos posteriores.



Redes Neuronales Artificiales (ANN)

Las ANN aplicadas están, en general, inspiradas en las redes neuronales biológicas,aunque poseen otras funcionalidades y estructuras de conexión distintas a las vistasdesde la perspectiva biológica. Las características principales de las ANN son lassiguientes:

1 Auto-Organización y Adaptabilidad : utilizan algoritmos de aprendizajeadaptativo y auto-organización, por lo que ofrecen mejores posibilidades deprocesado robusto y adaptativo.

2 *Procesado no Lineal : aumenta la capacidad de la red para aproximarfunciones, clasificar patrones y aumenta su inmunidad frente al ruido.

3 Procesado Paralelo: normalmente se usa un gran n´umero de nodos deprocesado, con alto nivel de interconectividad.



El elemento básico de computación (modelo de neurona) es un nodo o unidad.Recibe un input desde otras unidades o de una fuente externa de datos. Cada inputtiene un peso asociado w , que se va modificando en el llamado procesodeaprendizaje. Cada unidad aplica una funciónn dada f de la suma de los inputsponderadas mediante los pesos

yi =∑

jwijyj

El resultado puede servir como output de otras unidades.

Figure 3



Fases de Modelización

Fase de entrenamiento: se usa un conjunto de datos o patrones deentrenamiento para determinar los pesos (parámetros) que definen el modelode red neuronal. Se calculan de manera iterativa, de acuerdo con los valores delos valores de entrenamiento, con el objeto de minimizar el error cometidoentre la salida obtenida por la red neuronal y la salida deseada.Fase de Prueba : en la fase anterior, el modelo puede que se ajustedemasiado a las particularidades presentes en los patrones de entrenamiento,perdiendo su habilidad de generalizar su aprendizaje a casos nuevos(sobreajuste).



Para evitar el problema del sobreajuste, es aconsejable utilizar un segundogrupo de datos diferentes a los de entrenamiento, el grupo de validación, quepermita controlar el proceso de aprendizaje.Normalmente, los pesos óptimos se obtienen optimizando (minimizando)alguna función de energía.

Por ejemplo, un criterio muy utilizado en el llamado entrenamiento supervisado, queconsiste en minimizar el error cuadrático medio entre el valor de salida y el valorreal esperado.



Función de Activación

El valor de red, expresado por la función de base, u(w ; x), se transforma medianteuna función de activación no lineal. Las funciones de activación más comunes sonla función logística y la función tangente hiperbólica.

Función Logística:

f (x) = 11 + e−x

Función hiperbólica:

f (x) = ex − e−x

ex + e−x



Figure 4



Estructuras de conexión de atrás hacia delante

Una red neuronal se determina por las neuronas y la matriz de pesos. Se puedendefinir tres tipos de capas de neuronas:

la capa de entrada,la capa oculta yla capa de salida.

Entre dos capas de neuronas existe una red de pesos de conexión, que puede ser delos siguientes tipos:

hacia delante, hacia atrás, lateral y de retardo.



Tañano de las Redes Neuronales

En una red multicapa de propagación hacia delante, puede haber una o máscapas ocultas entre las capas de entrada y salida. El tamaño de las redesdepende del número de capas y del número de neuronas ocultas por capa.

El número de unidades ocultas está directamente relacionado con lascapacidades de la red. Para que el comportamiento de la red sea correcto, setiene quedeterminar apropiadamente el número de neuronas de la capa oculta.


Métodos de Regresión y Redes Neuronales




Relacionando conceptos

Los modelos de regresión estudian la relación entre una serie de variables,denominadas independientes xi , y otras variables dependientes o respuesta quese denotan como y : El objetivo es predecir los valores de y en función de losvalores de xi .

Un modelo lineal se puede implementar como una red neuronal simple: tieneuna unidad de sesgo, una unidad de input y una unidad de output. El input serefiere a una variable x , mientras que el sesgo siempre es una constante igual a1.



El output podría expresarse pues como:

y2 = y1w21 + 1.0w20

Figure 5



El modelo lineal como una ANNyp =

∑Ni=1 βixpi + �

Figure 6Xavi Barber (@umh1480 @XaviBarberUMH) Redes Neuronales 22 / 67


Capas Ocultas

Figure 7



Figure 8


Neural Networks y R

Neural Networks y R


Neural Networks y R

neuralnet

A mode de ejemplo vamos a utilizar está en el paquete MASS y se llama Boston. Enella se encuentran los precios de casas en una zona de dicha ciudad, y se quiererelacionar el precio con distintas variables.

set.seed(500)library(MASS)data

Neural Networks y R

crim: per capita crime rate by town.zn: proportion of residential land zoned for lots over 25,000 sq.ft.indus: proportion of non-retail business acres per town.chas: Charles River dummy variable (= 1 if tract bounds river; 0 otherwise).nox: nitrogen oxides concentration (parts per 10 million).rm: average number of rooms per dwelling.age: proportion of owner-occupied units built prior to 1940.dis: weighted mean of distances to five Boston employment centres.rad: index of accessibility to radial highways.tax: full-value property-tax rate per $ 10.000.ptratio: pupil-teacher ratio by town.black: 1000(Bk - 0.63)ˆ2 where Bk is the proportion of blacks by town.lstat: lower status of the population (percent).medv: median value of owner-occupied homes in $ 1000.

Our goal is to predict the median value of owner-occupied homes (medv) using all the other continuousvariables available.


Neural Networks y R

lm.fit |t|)## (Intercept) 43.469681 6.099347 7.127 5.50e-12 ***## crim -0.105439 0.057095 -1.847 0.065596 .## zn 0.044347 0.015974 2.776 0.005782 **## indus 0.024034 0.071107 0.338 0.735556## chas 2.596028 1.089369 2.383 0.017679 *## nox -22.336623 4.572254 -4.885 1.55e-06 ***## rm 3.538957 0.472374 7.492 5.15e-13 ***## age 0.016976 0.015088 1.125 0.261291## dis -1.570970 0.235280 -6.677 9.07e-11 ***## rad 0.400502 0.085475 4.686 3.94e-06 ***## tax -0.015165 0.004599 -3.297 0.001072 **## ptratio -1.147046 0.155702 -7.367 1.17e-12 ***## black 0.010338 0.003077 3.360 0.000862 ***## lstat -0.524957 0.056899 -9.226 < 2e-16 ***## ---## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#### (Dispersion parameter for gaussian family taken to be 23.26491)#### Null deviance: 33642 on 379 degrees of freedom## Residual deviance: 8515 on 366 degrees of freedom## AIC: 2290#### Number of Fisher Scoring iterations: 2


Neural Networks y R

pr.lm

Neural Networks y R

Ahora veamos cómo preparar los datos para realizar la red neuronal:

Normalizamos los datos

maxs

Neural Networks y R

No existe una regla definida que nos indique las “capas ocultas” que debe tener unaRed, así como el número de “neuronas”.

Tenemos 13 inputsVamos a coger 2 capas ocultas

5 neuronas para la primera capa (2/3 de los “inputs”)2 neuronas para la segunda capa (2/3 de los “inputs”)


Neural Networks y R

Dos consideraciones importantes:

En la librería neuralnet no se acepta poner y .. Tenemos que escribirprimero la formula y luego utilizarla dentro del comando.El argumento hidden acepta un vector con el número de neuronas para cadacapa oculta, mientras que el argumento linear.output se usa paraespecificar cómo queremos hacer la regresión: TRUE para regresión y FALSEpara clasificacación.


Neural Networks y R

library(neuralnet)n

Neural Networks y R

set.seed(1480)nn

Neural Networks y R

head(cbind(nn$response, nn$covariate))

## medv## 422 0.2044444444 0.078861183133 0 0.6466275660 0 0.6851851852 0.4684805518## 367 0.3755555556 0.041481791366 0 0.6466275660 0 0.6851851852 0.2686338379## 492 0.1911111111 0.001117456829 0 1.0000000000 0 0.4609053498 0.4640735773## 236 0.4222222222 0.003643143050 0 0.2104105572 0 0.2510288066 0.4838091588## 408 0.5088888889 0.134256447238 0 0.6466275660 0 0.5637860082 0.3922207319## 104 0.3177777778 0.002307410103 0 0.2969208211 0 0.2777777778 0.4935811458#### 422 0.9515962925 0.06774636488 1.0000000000 0.9141221374 0.8085106383## 367 0.9114315139 0.05662504888 1.0000000000 0.9141221374 0.8085106383## 492 0.9876416066 0.06715528922 0.1304347826 1.0000000000 0.7978723404## 236 0.6035015448 0.22936463913 0.3043478261 0.2290076336 0.5106382979## 408 1.0000000000 0.01414944212 1.0000000000 0.9141221374 0.8085106383## 104 0.8702368692 0.14414062145 0.1739130435 0.3759541985 0.8829787234#### 422 0.8060416562 0.3854856512## 367 0.7960814968 0.3385761589## 492 0.9828786121 0.4508830022## 236 0.9491905795 0.2524834437## 408 0.8365777397 0.2869757174## 104 0.9938726108 0.3231236203


Neural Networks y R

error.fct es una función diferenciable para el cálculo del error. Alternativamente,tenemos ‘sse’ y ‘ce’ los cuales usaremos como la suma del cuadrado de lso errores yla función cruzada de “entropía”.

nn$error.fct

## NULL


Neural Networks y R

Usaremos una función diferrenciable act.fct para suavizar las resultdos delproducto cruzado de la covariable so neuronas y los pesos. Adicionamente “logistic”y “atanh” son las siglas para la función logística y las tangente hiperbólica.

nn$act.fct

## function (x)## {## 1/(1 + exp(-x))## }## ## attr(,"type")## [1] "logistic"


Neural Networks y R

Si no aplicamos acr.fct al output de las neuronas el que sea una combinaciónlineal=TRUE, por defecto es FALSE.

nn$linear.output

## [1] TRUE


Neural Networks y R

Valores estimados según la red neuronal para nuestros datos, pero CUIDADO, estánestandarizado

head(pr.nn$net.result)


Neural Networks y R

Volviendo los datos a la escala orginal (recordar que hemos “escalado” los datos):

pr.nn_noesta

Neural Networks y R

Estudiando el “peso” de las covariables

weights, a list containing the fitted weights of the neural network for eachreplication.startweights , a list containing the starting weights for each replication.generalized.weights , a list containing the generalized weights of the neuralnetwork for each replication.

The generalized weight expresses the effect of each covariate xi and thus has ananalogous interpretation as the ith regression parameter in regression models.

However, the generalized weight depends on all other covariates. Its distributionindicates whether the effect of the covariate is linear since a small variance suggestsa linear effect (Intrator and Intrator,2001).


Neural Networks y R

nn$weightsnn$startweightsnn$generalized.weights

head(nn$weights)[[1]]

## [[1]]## [,1] [,2] [,3] [,4]## [1,] 10.404004008 0.30155847594 -0.431649943198 -0.16645952652## [2,] 9.562567037 3.90509120100 -1.351173695096 -4.53992636238## [3,] 7.908661679 87.33084382784 0.007648117799 0.30097295207## [4,] 8.932370408 -1.30599545221 -0.236768979443 0.81000974705## [5,] 4.935489861 0.36519368946 -0.267012706875 11.44483263709## [6,] 8.245878605 6.40510150820 -0.304627168250 0.51769629635## [7,] 10.121266185 0.02035836275 5.276190448040 1.95464590359## [8,] 8.835886842 -2.28221894784 -0.446782947654 -1.57421227407## [9,] 7.346144720 20.18700441636 -1.250830326437 0.06489203261## [10,] 9.401024309 -0.88659482890 -0.864208401435 5.93020707160## [11,] 5.048018558 -1.67284062152 -2.080868311135 -0.50967979137## [12,] 8.770071656 0.15871538688 -1.064340969678 -0.66338330846## [13,] 7.134547073 0.02049836217 -1.282407647947 1.03351663520## [14,] 8.349296671 3.65814774846 -3.390089730921 0.98381528661## [,5]## [1,] 2.3535980472## [2,] -14.3993809593## [3,] 2.8933949463## [4,] -1.2436175138## [5,] -2.5230866455## [6,] 5.8508313118## [7,] -2.7495515235## [8,] -4.5633498053## [9,] 38.9931506750## [10,] 0.1213566907## [11,] -4.9130531342## [12,] 2.1479308460## [13,] 3.3764302140## [14,] -5.2377936012#### [[2]]## [,1] [,2] [,3]## [1,] 0.6213662933 -0.2030652955 0.3159054889## [2,] -0.5336160666 -1.0116250730 -0.3642337263## [3,] -0.7073147322 0.3462393330 -3.3496382469## [4,] 0.8750650749 -0.7014690707 2.7497947422## [5,] 0.9201045567 -0.4427591711 0.2922708272## [6,] 0.3630024807 -0.6476805529 -0.9430866930#### [[3]]## [,1]## [1,] -0.6609425676## [2,] 1.8307651390## [3,] -1.9811802884## [4,] 2.3374208364

head(nn$startweights)

## [[1]]## [[1]][[1]]## [,1] [,2] [,3] [,4]## [1,] 1.08837294916 0.23595029754 0.007699810867 0.3855380738## [2,] 0.46638787881 -0.43730379336 -0.526878377109 0.9322623097## [3,] -0.43935826932 -0.75944756513 -1.210292486823 -0.5674681797## [4,] 1.35155580803 -0.29398025699 0.033434452297 0.6734642456## [5,] -0.04018725917 0.96421906565 -0.957127779329 0.4950998332## [6,] 1.36383435933 0.99898092262 -0.785918473914 0.6166241781## [7,] 0.08428280967 -0.90424853334 -0.585474460383 -0.3937015073## [8,] 1.40268774096 -1.18221386572 0.168159253131 -0.5989293688## [9,] 0.14428844282 2.44751340234 -0.501149843284 0.9251040046## [10,] 1.30943156326 -0.02366286310 -0.860142607397 1.3314553390## [11,] -1.67384501435 -1.03903909291 -0.249181303141 -0.7447518211## [12,] 0.21892123641 0.19598005930 -0.463522312278 -0.4772549554## [13,] -0.45793420432 0.08269719239 0.569349072996 1.3999549624## [14,] -0.72884002002 -0.48130934395 0.313285542423 0.5212344004## [,5]## [1,] 2.0687925651## [2,] -1.6817795967## [3,] -1.7467495108## [4,] 0.2024800418## [5,] -1.5605732432## [6,] -0.5795710144## [7,] 0.6048695729## [8,] -1.2192404006## [9,] -0.9328411204## [10,] 0.4809397956## [11,] 0.5922545851## [12,] -0.6547226620## [13,] 0.5285855454## [14,] 1.4062331560#### [[1]][[2]]## [,1] [,2] [,3]## [1,] 0.3834413325 0.11283256754 -0.49024666856## [2,] -0.7715410274 -0.69572720996 -1.17038588376## [3,] -0.8497227386 0.55663202298 -2.87166405592## [4,] 0.8697751324 0.57179448945 0.62148016571## [5,] 0.6750605455 -0.38376114697 0.06504758338## [6,] -0.0677647042 -0.02425139078 -0.40579195910#### [[1]][[3]]## [,1]## [1,] -0.9774594216## [2,] 1.3814036265## [3,] -2.1343966539## [4,] -1.0286057276

head(nn$generalized.weights)

## [[1]]## [,1] [,2] [,3] [,4]## 422 -5.9627744839 -23.0364350096273 -0.0574470963363 -0.857794334033## 367 -4.8821452862 -25.6946804061242 0.0837958659042 -0.662460191859## 492 -8.9610560272 -93.6778812261574 1.6289443675283 5.239061279339## 236 -0.9836181755 -0.9542575029156 0.0383275808539 1.255733140753## 408 -5.4223089118 -116.3101488121751 1.7306176537971 -0.381116701505## 104 -2.3742683378 -8.3492978854678 0.3850245977973 4.201961204607## 257 -2.6355966754 0.0676652464511 -0.1541938109732 1.844074208422## 462 -0.7924025962 -4.9148362648389 0.0267767188287 -0.082013925680## 413 -2.3378615063 -44.7525701177930 0.7256437581367 0.628072312008## 354 -1.3048676011 0.0374626402519 -0.0532319633377 1.090626093470## 140 -7.2005975303 -26.3884728427886 0.3672933511543 3.802096711111## 442 -6.1088809823 -7.5702509216173 -0.3589981737883 -0.958982057432## 378 -3.4476517668 -68.7951673700606 0.9995510303019 -0.280453575318## 82 -1.9723457779 0.0895168307578 0.1113749835435 3.123176443187## 361 -0.4557289159 -4.8236552173589 0.0507403239634 -0.016737015053## 125 -3.0219538742 -12.0285920124529 0.3919226270368 4.093760666642## 250 -0.8965901813 0.0247523711848 -0.0423426948930 0.705055702086## 303 -0.9197828463 0.0215157373544 -0.0660503269434 0.549471040672## 297 -1.4488924399 0.0001144522079 0.0038595190995 1.686157370192## 124 -4.2570255076 -6.8634578401040 0.2185343178671 3.996523351519## 38 -1.3804299719 -0.3499958819122 0.1020106679834 2.300045154912## 393 -4.3027983768 -21.0294836591950 0.0556520814192 -0.471675229459## 19 -1.9771546684 -0.1433248821732 0.2545815535733 4.185752966327## 261 -1.9058957214 0.0550038580338 -0.1095338504963 1.327608008336## 130 -4.1932711118 -13.1046099009539 0.4010098561183 4.556086427862## 392 -3.3672406757 -7.0299498551895 -0.1446617896316 -0.516678044259## 334 -1.4127524038 0.0531188618016 0.0349971449397 1.892289401461## 429 -3.1408780543 -25.1387469114883 0.2166727002589 -0.264714805699## 20 -2.2463086410 -0.4683863358817 0.3144818868436 4.896936721669## 63 -1.0524074106 0.0423832560466 0.0280415316541 1.425202203956## 33 -2.4526068527 0.0425405969690 0.3926235914739 5.812225561195## 484 -0.1202133800 -1.7040915153776 0.0216667052607 0.013436346230## 293 -1.5221722204 0.0457925994513 -0.0499001499599 1.366009509205## 47 -1.4597897126 0.0353701642101 0.1349929288897 2.708818092873## 178 -1.9497119748 -3.0607698669631 0.1134329231220 2.577319041122## 397 -5.1718000243 -39.5721771091634 0.3131716123051 -0.689986808666## 388 -1.0477426440 -14.5320028082960 0.2227780711182 0.269578174792## 478 -3.1556268108 -34.2648493095712 0.4041485979229 -0.147584364409## 435 -3.9151799714 -19.0732042014404 0.0539480680337 -0.232714539274## 22 -2.5262113899 -0.3647201044616 0.3688052781318 5.650456455841## 91 -2.6393771576 -3.1962393939815 0.2207013132518 4.163562634551## 197 -2.2729917915 0.0637565283638 -0.1014804094647 1.832505537365## 387 -1.5012195505 -24.7355891210899 0.3943439246704 0.419212309542## 225 -8.3055551159 -72.2262613970771 0.5702107003334 1.603605584940## 395 -5.2167381899 -34.2371269531819 0.2182547690679 -0.642953447310## 272 -1.5386332454 0.0550646102217 0.0007506785625 1.774286497345## 265 -2.1702431789 0.0732121807322 -0.1000640537256 1.678019243345## 215 -0.6488647496 -0.0237366513056 0.0896525422471 1.423882199610## 110 -2.5464789679 -9.9449503664454 0.3974549278348 4.129756095862## 109 -3.0102599252 -18.0773247656404 0.4890975781859 3.961390407454## 106 -4.0950684895 -22.2884938619531 0.5756000557284 4.626577184307## 450 -6.1561046621 -11.1375415486493 -0.2962528278350 -0.937365181848## 213 -0.2090466103 -0.3945403826866 -0.0270287036588 -0.039285213313## 32 -2.9602289431 -0.3748209533245 0.4283772303107 6.601137273820## 88 -2.2196169114 -1.2791488029903 0.1795016034296 3.695071817971## 294 -1.1077013212 -0.0096870906720 -0.0084008140277 1.200504918782## 36 -1.8048413369 -1.9830855454066 0.1902992102404 3.183575371083## 365 -4.0732458231 -37.5912830871310 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Redes Neuronales - Introduciondonos en la Inteligencia...

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