x1<-rnorm(30,1,1)
x2<-rnorm(30,0,1)
x3<-rnorm(30,2,1)
x=matrix(cbind(x1,x2,x3),30,3)
x
## [,1] [,2] [,3]
## [1,] 1.5997300 -0.14900413 2.2127523
## [2,] -0.4916623 0.54031892 1.7391625
## [3,] 1.0528734 -0.48156364 2.9380617
## [4,] 0.1819714 -1.82281212 1.8736057
## [5,] 1.1760739 0.53073197 1.9159576
## [6,] -0.1316264 -0.05462993 1.5665069
## [7,] 1.1671482 -1.15890113 1.7268290
## [8,] 0.6477115 -1.84077690 1.7398021
## [9,] 1.4519454 1.03899388 0.8114545
## [10,] 0.2241078 0.07457158 0.9634745
## [11,] 1.0651405 -1.66035380 2.8129895
## [12,] 0.9124404 -0.29711037 2.9669679
## [13,] 1.1393169 -0.98410188 2.3712689
## [14,] -1.1448898 -1.52090720 3.7715338
## [15,] -0.8203004 0.43725214 2.0015467
## [16,] -0.3983989 0.07039490 1.2979282
## [17,] 2.8890790 0.08679356 2.7542641
## [18,] 1.7797558 -1.57220671 2.6134938
## [19,] -0.5450273 -1.41051424 1.8008150
## [20,] 1.7681414 -0.30389543 1.4777686
## [21,] 0.6652090 -1.24879886 1.1752352
## [22,] 1.0976406 1.08071404 4.0048361
## [23,] 3.9534878 0.25520562 2.6646543
## [24,] 1.1311492 -0.39968724 2.1521996
## [25,] 1.0805652 0.33565133 2.9960124
## [26,] 0.6319068 -0.40381400 1.3222619
## [27,] -0.4180250 -1.67538019 2.9970021
## [28,] 1.6319200 -0.36491594 3.6474563
## [29,] 0.4941100 -0.86967207 2.3364463
## [30,] 1.4055086 0.63041387 3.3806660
n<-30
#Kovarian
cov(x)
## [,1] [,2] [,3]
## [1,] 1.1602302 0.23318829 0.15707593
## [2,] 0.2331883 0.77318296 -0.01375871
## [3,] 0.1570759 -0.01375871 0.71202872
cor(x)
## [,1] [,2] [,3]
## [1,] 1.0000000 0.24620310 0.17281793
## [2,] 0.2462031 1.00000000 -0.01854334
## [3,] 0.1728179 -0.01854334 1.00000000
#lihat bahwa nilainya kecil, coba kita buat manual
#vektor satu
satu<-matrix(c(rep(1,n)))
satu
## [,1]
## [1,] 1
## [2,] 1
## [3,] 1
## [4,] 1
## [5,] 1
## [6,] 1
## [7,] 1
## [8,] 1
## [9,] 1
## [10,] 1
## [11,] 1
## [12,] 1
## [13,] 1
## [14,] 1
## [15,] 1
## [16,] 1
## [17,] 1
## [18,] 1
## [19,] 1
## [20,] 1
## [21,] 1
## [22,] 1
## [23,] 1
## [24,] 1
## [25,] 1
## [26,] 1
## [27,] 1
## [28,] 1
## [29,] 1
## [30,] 1
#vektor rata-rata
vxbar<-1/n*t(x)%*%satu
vxbar
## [,1]
## [1,] 0.8399001
## [2,] -0.4379335
## [3,] 2.2677651
#matriks rata-rata
xbar=1/n*satu%*%t(satu)%*%x #30 x 3
xbar
## [,1] [,2] [,3]
## [1,] 0.8399001 -0.4379335 2.267765
## [2,] 0.8399001 -0.4379335 2.267765
## [3,] 0.8399001 -0.4379335 2.267765
## [4,] 0.8399001 -0.4379335 2.267765
## [5,] 0.8399001 -0.4379335 2.267765
## [6,] 0.8399001 -0.4379335 2.267765
## [7,] 0.8399001 -0.4379335 2.267765
## [8,] 0.8399001 -0.4379335 2.267765
## [9,] 0.8399001 -0.4379335 2.267765
## [10,] 0.8399001 -0.4379335 2.267765
## [11,] 0.8399001 -0.4379335 2.267765
## [12,] 0.8399001 -0.4379335 2.267765
## [13,] 0.8399001 -0.4379335 2.267765
## [14,] 0.8399001 -0.4379335 2.267765
## [15,] 0.8399001 -0.4379335 2.267765
## [16,] 0.8399001 -0.4379335 2.267765
## [17,] 0.8399001 -0.4379335 2.267765
## [18,] 0.8399001 -0.4379335 2.267765
## [19,] 0.8399001 -0.4379335 2.267765
## [20,] 0.8399001 -0.4379335 2.267765
## [21,] 0.8399001 -0.4379335 2.267765
## [22,] 0.8399001 -0.4379335 2.267765
## [23,] 0.8399001 -0.4379335 2.267765
## [24,] 0.8399001 -0.4379335 2.267765
## [25,] 0.8399001 -0.4379335 2.267765
## [26,] 0.8399001 -0.4379335 2.267765
## [27,] 0.8399001 -0.4379335 2.267765
## [28,] 0.8399001 -0.4379335 2.267765
## [29,] 0.8399001 -0.4379335 2.267765
## [30,] 0.8399001 -0.4379335 2.267765
# matriks deviasi
D<-x-xbar
D
## [,1] [,2] [,3]
## [1,] 0.7598299 0.28892934 -0.05501277
## [2,] -1.3315624 0.97825238 -0.52860257
## [3,] 0.2129733 -0.04363017 0.67029659
## [4,] -0.6579287 -1.38487865 -0.39415945
## [5,] 0.3361738 0.96866543 -0.35180756
## [6,] -0.9715265 0.38330354 -0.70125817
## [7,] 0.3272481 -0.72096766 -0.54093613
## [8,] -0.1921886 -1.40284344 -0.52796300
## [9,] 0.6120453 1.47692735 -1.45631062
## [10,] -0.6157923 0.51250504 -1.30429064
## [11,] 0.2252404 -1.22242034 0.54522434
## [12,] 0.0725403 0.14082310 0.69920277
## [13,] 0.2994168 -0.54616841 0.10350375
## [14,] -1.9847899 -1.08297373 1.50376866
## [15,] -1.6602005 0.87518561 -0.26621845
## [16,] -1.2382990 0.50832837 -0.96983693
## [17,] 2.0491789 0.52472703 0.48649903
## [18,] 0.9398557 -1.13427324 0.34572865
## [19,] -1.3849274 -0.97258078 -0.46695008
## [20,] 0.9282414 0.13403804 -0.78999647
## [21,] -0.1746911 -0.81086539 -1.09252991
## [22,] 0.2577405 1.51864751 1.73707098
## [23,] 3.1135877 0.69313908 0.39688915
## [24,] 0.2912491 0.03824622 -0.11556554
## [25,] 0.2406652 0.77358479 0.72824732
## [26,] -0.2079933 0.03411946 -0.94550321
## [27,] -1.2579251 -1.23744672 0.72923701
## [28,] 0.7920200 0.07301752 1.37969119
## [29,] -0.3457900 -0.43173860 0.06868116
## [30,] 0.5656085 1.06834734 1.11290090
s=1/(n-1)*t(D)%*%D
s
## [,1] [,2] [,3]
## [1,] 1.1602302 0.23318829 0.15707593
## [2,] 0.2331883 0.77318296 -0.01375871
## [3,] 0.1570759 -0.01375871 0.71202872
#cek dengan cov
Dev=diag(sqrt(diag(s)))
Dev
## [,1] [,2] [,3]
## [1,] 1.07714 0.0000000 0.0000000
## [2,] 0.00000 0.8793082 0.0000000
## [3,] 0.00000 0.0000000 0.8438179
R=solve(Dev)%*%s%*%solve(Dev)
R
## [,1] [,2] [,3]
## [1,] 1.0000000 0.24620310 0.17281793
## [2,] 0.2462031 1.00000000 -0.01854334
## [3,] 0.1728179 -0.01854334 1.00000000
#Kombinasi Linier
A=matrix(c(1,2,3,4,5,6),2,3)
A
## [,1] [,2] [,3]
## [1,] 1 3 5
## [2,] 2 4 6
#Ambil baris 1 matriks A
a=A[1,]
a
## [1] 1 3 5
b=A[2,]
b
## [1] 2 4 6
#Dicari satu-satu
rata1=t(a)%*%vxbar
rata2=t(b)%*%vxbar
rata1
## [,1]
## [1,] 10.86493
rata2
## [,1]
## [1,] 13.53466
RATA=A%*%vxbar
RATA
## [,1]
## [1,] 10.86493
## [2,] 13.53466
#Varians
cov1=t(a)%*%s%*%a
cov2=t(b)%*%s%*%b
#Kovarians
cov3=t(a)%*%s%*%b
cov4=t(b)%*%s%*%a
cov1
## [,1]
## [1,] 28.47672
cov2
## [,1]
## [1,] 49.4853
cov3
## [,1]
## [1,] 37.28178
cov4
## [,1]
## [1,] 37.28178
COV=A%*%s%*%t(A)
COV
## [,1] [,2]
## [1,] 28.47672 37.28178
## [2,] 37.28178 49.48530