Creacion de matrices

Matriz Y

#Generacion de la matriz Y
Y <- matrix(c( 30, 20, 36, 24, 40),
            nrow = 5, 
            ncol = 1,
            byrow = TRUE)
colnames(Y)<- c("Y") 
print(Y)
##       Y
## [1,] 30
## [2,] 20
## [3,] 36
## [4,] 24
## [5,] 40

Matriz X

X  <- cbind(rep(1,5), matrix( data= c(4, 10 , 3, 8, 6, 11, 4, 9, 8, 12), nrow = 5, ncol = 2, byrow = TRUE))
colnames(X) <- c("Cte", "x1", "x2") 
print(X)
##      Cte x1 x2
## [1,]   1  4 10
## [2,]   1  3  8
## [3,]   1  6 11
## [4,]   1  4  9
## [5,]   1  8 12

Producto de matrices Calculo de X’X (Sigma matriz)

#Creando sigma matriz o, X'X (X trans * X)

matriz_Sigma <- t(X)%*%X

print(matriz_Sigma)
##     Cte  x1  x2
## Cte   5  25  50
## x1   25 141 262
## x2   50 262 510

La siguiente operacion obtiene la matrix X’Y

XY <- t(X)%*%Y
print(XY)
##        Y
## Cte  150
## x1   812
## x2  1552

Calculando la inversa de matrices

XX_inv <- solve(matriz_Sigma)
print(XX_inv)
##        Cte     x1    x2
## Cte 40.825  4.375 -6.25
## x1   4.375  0.625 -0.75
## x2  -6.250 -0.750  1.00

Obtencion del estimador MCO

Beta <- XX_inv%*%XY
colnames(Beta) <- c("Parametros")
print(Beta)
##     Parametros
## Cte     -23.75
## x1       -0.25
## x2        5.50

Obtencion de autovalores y autovectores Se usara el comando eigen()

autovalores <- eigen(matriz_Sigma)
print(autovalores)
## eigen() decomposition
## $values
## [1] 650.78185037   5.19448432   0.02366531
## 
## $vectors
##             [,1]       [,2]       [,3]
## [1,] -0.08623239  0.1629390  0.9828606
## [2,] -0.45874789 -0.8822205  0.1060061
## [3,] -0.88437229  0.4417441 -0.1508239

Muestra solo autovalores

print(autovalores$values)
## [1] 650.78185037   5.19448432   0.02366531