Simulación MEL
MVelandia
2022-08-02
Paso 1
Para la simulación vamos a generar un conjunto de datos basandonos en un modelo con parametros previamente definidos: \(Y_i\)=5 +(3\(X_i\))+\(\epsilon_i\), en donde podemos notar que \(\beta_0\)=5,\(\beta_1\)=3 y \(\epsilon_i \sim{\sf Norm}(0,\delta^2)\), \(i\)=1,2,3,…,\(n\) e independientes
x=runif(30,1,30) ## Valores iniciales arbitrarios para X
y = 5+(3*x) + rnorm(n = 30,mean = 0,sd = 10 ) ##Valores de Y de acuerdo al modelo
plot(x,y, xlim = c(0,30), ylim = c(0,100), las=1, pch=19, col="#b0394a")
datos=data.frame(x,y)
desv=10
head(datos,10)## x y
## 1 6.997889 17.832103
## 2 14.312444 28.048308
## 3 7.037918 31.477798
## 4 2.164430 13.394175
## 5 2.054613 6.461188
## 6 4.236070 25.889018
## 7 28.864193 88.426265
## 8 15.984191 39.062233
## 9 20.346960 70.132736
## 10 23.941139 83.327567Paso 2
Ahora vamos a asumir valores para la estimación de \(\beta_0\)=4 y \(\beta_1\)=3.5, posteriormente graficamos la linea y obtenemos el valor de MEL.
b0=4 # intercepto estimado
b1=2.5 # pendiente estimada
x1=1:30
y1 = b0 + b1 * x1
plot(x,y, xlim = c(0,30), ylim = c(0,100), las=1,pch=19 ,col=3)
lines(x1, y1 , col = 2, type = "l")
grid()
Paso 3
Se realiza el Paso 2 para otros posibles valores de los parametros y se guarda el valor de la probabilidad de la ecuación L de verosimilitud, con \(\delta\)=10 constante
Ln(L)=\(\frac{-n}{2}Ln(2\pi)-\frac{n}{2}Ln(\delta^2)-\frac{1}{2\delta^2}\sum_{i=1}^n(y_i-\beta_0-\beta_1x_i)^2\)
beta0_est= seq(3,7,0.2) # rango de valores al rededor de beta0 = 5
beta1_est=seq(1,5,0.2) # rango de valores al rededor de beta1 = 3
betas=expand.grid(beta0_est,beta1_est)
names(betas)=c("beta0_est","beta1_est")
L=array(NA,dim(betas)[1])
plot(x,y, ylim = c(0,100))
n=30
parte1=(-n/2)*log(2*pi)-(n/2)*log(desv^2)
parte2=(-1/2*desv^2)
for(i in 1:dim(betas)[1]){
y_est=betas$beta0_est[i] + (betas$beta1_est[i]*x)
lines(x,y_est,col=2)
parte3=(y_est-betas$beta0_est[i]-betas$beta1_est[i]*x)^2
parte3=sum(y_est-y)^2
res=(parte1+parte2*parte3)
L[i]=res
} 
resultados=data.frame(betas,L)Paso 4
Graficar e interpretar la relación entre los posibles Parametros (betas) y la MEL.
library(plotly)
plot_ly(x=resultados$beta0_est,
y=resultados$beta1_est,
z=resultados$L,
size=.5)Conclusión
Como observamos en la gráfica anterior el logaritmo natural de la ecuación de verosimilitud da una curva inversa donde los valores de \(\beta_0\) y \(\beta_1\) se encuentran cuando en el valor máximo de la función L
df_ordenado=resultados[rev(order(resultados$L)),]
df_ordenado## beta0_est beta1_est L
## 220 4.8 3.0 -1.408195e+02
## 221 5.0 3.0 -1.376859e+03
## 219 4.6 3.0 -2.504780e+03
## 222 5.2 3.0 -6.212899e+03
## 218 4.4 3.0 -8.468741e+03
## 223 5.4 3.0 -1.464894e+04
## 217 4.2 3.0 -1.803270e+04
## 224 5.6 3.0 -2.668498e+04
## 210 7.0 2.8 -3.013462e+04
## 216 4.0 3.0 -3.119666e+04
## 225 5.8 3.0 -4.232102e+04
## 209 6.8 2.8 -4.664086e+04
## 215 3.8 3.0 -4.796062e+04
## 232 3.0 3.2 -6.006032e+04
## 226 6.0 3.0 -6.155706e+04
## 208 6.6 2.8 -6.674710e+04
## 214 3.6 3.0 -6.832458e+04
## 233 3.2 3.2 -8.263864e+04
## 227 6.2 3.0 -8.439310e+04
## 207 6.4 2.8 -9.045334e+04
## 213 3.4 3.0 -9.228854e+04
## 234 3.4 3.2 -1.088170e+05
## 228 6.4 3.0 -1.108291e+05
## 206 6.2 2.8 -1.177596e+05
## 212 3.2 3.0 -1.198525e+05
## 235 3.6 3.2 -1.385953e+05
## 229 6.6 3.0 -1.408652e+05
## 205 6.0 2.8 -1.486658e+05
## 211 3.0 3.0 -1.510165e+05
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## 204 5.8 2.8 -1.831721e+05
## 237 4.0 3.2 -2.089519e+05
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## 203 5.6 2.8 -2.212783e+05
## 238 4.2 3.2 -2.495302e+05
## 202 5.4 2.8 -2.629845e+05
## 239 4.4 3.2 -2.937085e+05
## 201 5.2 2.8 -3.082908e+05
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## 200 5.0 2.8 -3.571970e+05
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