Rangkuman Praktikum 1 ADM 1
Ester Nibenia Zendrato
2025-09-10
1 OPERASI MATRIKS
1.1 Penjumlahan
Penjumlahan matriks adalah operasi penjumlahan dua matriks dengan menjumlahkan komponen-komponennya yang seletak. Syaratnya adalah memiliki ordo yang sama.
Contoh : \[ \boldsymbol A = \begin{bmatrix} p & q \\ r & s \\ \end{bmatrix} \boldsymbol+\begin{bmatrix} v & w \\ x & y \\ \end{bmatrix} = \begin{bmatrix} p+v & q+w \\ r+x & s+y \\ \end{bmatrix} \]
Syntax R untuk penjumlahan matriks:
#input matriks
X <- matrix(c(1,2,3,
4,5,6,
7,8,9), nrow=3, byrow=TRUE);X## [,1] [,2] [,3]
## [1,] 1 2 3
## [2,] 4 5 6
## [3,] 7 8 9Y <- matrix(c(10,11,12,
13,14,15,
16,17,18), nrow=3, byrow=TRUE);Y## [,1] [,2] [,3]
## [1,] 10 11 12
## [2,] 13 14 15
## [3,] 16 17 18#penjumlahan
X + Y## [,1] [,2] [,3]
## [1,] 11 13 15
## [2,] 17 19 21
## [3,] 23 25 271.2 Pengurangan
Konsep pengurangan pada matriks sama seperti penjumlahan,yaitu mengurangkan elemen yang letaknya sama atau seletak. Salah satu syarat yang harus dipenuhi juga sama, yaitu memiliki ordo yang sama.
Contoh: \[ \boldsymbol A = \begin{bmatrix} p & q \\ r & s \\ \end{bmatrix} \boldsymbol-\begin{bmatrix} v & w \\ x & y \\ \end{bmatrix} = \begin{bmatrix} p-v & q-w \\ r-x & s-y \\ \end{bmatrix} \]
Syntax R untuk pengurangan matriks:
#input matriks
X <- matrix(c(1,2,3,
4,5,6,
7,8,9), nrow=3, byrow=TRUE);X## [,1] [,2] [,3]
## [1,] 1 2 3
## [2,] 4 5 6
## [3,] 7 8 9Y <- matrix(c(10,11,12,
13,14,15,
16,17,18), nrow=3, byrow=TRUE);Y## [,1] [,2] [,3]
## [1,] 10 11 12
## [2,] 13 14 15
## [3,] 16 17 18#pengurangan
X - Y## [,1] [,2] [,3]
## [1,] -9 -9 -9
## [2,] -9 -9 -9
## [3,] -9 -9 -91.3 Perkalian
- Perkalian skalar dengan matriks
Contoh: \[ \boldsymbol k\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} \boldsymbol=\begin{bmatrix} ka & kb \\ kc & kd \\ \end{bmatrix}\]
Syntax R untuk perkalian skalar dengan matriks:
#input matriks
k <- 2
Y <- matrix(c(10,11,12,
13,14,15,
16,17,18), nrow=3, byrow=TRUE);Y## [,1] [,2] [,3]
## [1,] 10 11 12
## [2,] 13 14 15
## [3,] 16 17 18#Perkalian
k*Y## [,1] [,2] [,3]
## [1,] 20 22 24
## [2,] 26 28 30
## [3,] 32 34 36- Perkalian antar matriks Perkalian matriks adalah nilai pada matriks yang bisa dihasilkan dengan cara dikalikan-nya tiap baris dengan setiap kolom yang memiliki jumlah baris yang sama.
Contoh:
Misalkan ada matriks:
\[ A = \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1q} \\ a_{21} & a_{22} & \dots & a_{2q} \\ \vdots & \vdots & \ddots & \vdots \\ a_{p1} & a_{p2} & \dots & a_{pq} \end{bmatrix}_{p \times q}, \quad B = \begin{bmatrix} b_{11} & b_{12} & \dots & b_{1n} \\ b_{21} & b_{22} & \dots & b_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ b_{q1} & b_{q2} & \dots & b_{qn} \end{bmatrix}_{q \times n} \]
Maka hasil perkalian \(AB\) adalah:
\[ AB = \begin{bmatrix} \sum_{k=1}^{q} a_{1k} b_{k1} & \sum_{k=1}^{q} a_{1k} b_{k2} & \dots & \sum_{k=1}^{q} a_{1k} b_{kn} \\ \sum_{k=1}^{q} a_{2k} b_{k1} & \sum_{k=1}^{q} a_{2k} b_{k2} & \dots & \sum_{k=1}^{q} a_{2k} b_{kn} \\ \vdots & \vdots & \ddots & \vdots \\ \sum_{k=1}^{q} a_{pk} b_{k1} & \sum_{k=1}^{q} a_{pk} b_{k2} & \dots & \sum_{k=1}^{q} a_{pk} b_{kn} \end{bmatrix}_{p \times n} \]
1.4 Transpose
Transpose dari sebuah matriks adalah matriks baru yang diperoleh dengan menukar baris menjadi kolom, atau sebaliknya.
Contoh: \[ \boldsymbol A =\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} \boldsymbol A'=\begin{bmatrix} a & c \\ b & d \\ \end{bmatrix}\]
Syntax R untuk transpose:
#input matriks
X <- matrix(c(1,2,3,
4,5,6,
7,8,9), nrow=3, byrow=TRUE);X## [,1] [,2] [,3]
## [1,] 1 2 3
## [2,] 4 5 6
## [3,] 7 8 9Y <- matrix(c(10,11,12,
13,14,15,
16,17,18), nrow=3, byrow=TRUE);Y## [,1] [,2] [,3]
## [1,] 10 11 12
## [2,] 13 14 15
## [3,] 16 17 18#transpose matriks
transX = t(X); transX## [,1] [,2] [,3]
## [1,] 1 4 7
## [2,] 2 5 8
## [3,] 3 6 9transY = t(Y); transY## [,1] [,2] [,3]
## [1,] 10 13 16
## [2,] 11 14 17
## [3,] 12 15 181.5 Invers
Invers matriks adalah matriks kebalikan (dilambangkan A⁻¹) yang, jika dikalikan dengan matriks aslinya, akan menghasilkan matriks identitas (I).
Contoh: \[ \boldsymbol A =\begin{bmatrix} x & w \\ y & z \\ \end{bmatrix}\]
\[ \boldsymbol{A^{-1}} = \frac{1}{\det{A}}\times adj({A}) \]
Syntax R untuk invers matriks:
#input matriks
X <- matrix(c(4, 2,
3, 1), nrow = 2, byrow = TRUE)
#invers matriks
inv_X = solve(X); inv_X## [,1] [,2]
## [1,] -0.5 1
## [2,] 1.5 -21.6 Determinan
Determinan adalah nilai skalar unik yang dapat dihitung dari unsur-unsur matriks persegi dan dilambangkan dengan det(A) atau |A|
Contoh: \[ \boldsymbol A =\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}\]
\[ \ det(A) = ab - dc\]
Syntax R untuk determinan matriks:
#input matriks
X <- matrix(c(1,2,3,
4,5,6,
7,8,9), nrow=3, byrow=TRUE);X## [,1] [,2] [,3]
## [1,] 1 2 3
## [2,] 4 5 6
## [3,] 7 8 9Y <- matrix(c(10,11,12,
13,14,15,
16,17,18), nrow=3, byrow=TRUE);Y## [,1] [,2] [,3]
## [1,] 10 11 12
## [2,] 13 14 15
## [3,] 16 17 18#determinan matriks
det(X)## [1] 6.661338e-16det(Y)## [1] 01.7 Cara memanggil komponen matriks di R:
A <- matrix(21:40, nrow=4, ncol=5)
A## [,1] [,2] [,3] [,4] [,5]
## [1,] 21 25 29 33 37
## [2,] 22 26 30 34 38
## [3,] 23 27 31 35 39
## [4,] 24 28 32 36 40A[,2] # Kolom 2## [1] 25 26 27 28A[3,] # Baris 3## [1] 23 27 31 35 39A[3,2] # Sel(3, 2)## [1] 27A[c(1,3),2] # Sel(1,2) dan sel(3,2)## [1] 25 27A[,1:3] # kolom(1,2,3)## [,1] [,2] [,3]
## [1,] 21 25 29
## [2,] 22 26 30
## [3,] 23 27 31
## [4,] 24 28 32A[2:4,] # baris(2,3,4)## [,1] [,2] [,3] [,4] [,5]
## [1,] 22 26 30 34 38
## [2,] 23 27 31 35 39
## [3,] 24 28 32 36 402 EIGEN VALUE
2.1 Eigen value
Eigenvalue adalah skalar (angka) yang menunjukkan seberapa besar eigenvector diperbesar atau diperkecil ketika matriks A mengalikan vektor itu.
Contoh: \[ \boldsymbol A = \begin{bmatrix} 2 & 0 \\ 1 & 3 \end{bmatrix} \]
\[ \det(A - \lambda I) = \det \begin{bmatrix} 2-\lambda & 0 \\ 1 & 3-\lambda \end{bmatrix} \]
\[ \det(A - \lambda I) = (2-\lambda)(3-\lambda) - (1)(0) \]
\[ \det(A - \lambda I) = \lambda^2 -5\lambda +6 \]
\[ \det(A - \lambda I) = (\lambda - 2)(\lambda - 3) \]
Dari sini didapat: \[ \lambda_1 = 2, \quad \lambda_2 = 3 \]
2.2 Eigen vector
Eigenvector adalah vektor non-nol yang arahnya tidak berubah ketika dikalikan dengan matriks A.
Contoh: \[ \text{Matriks: } \boldsymbol A = \begin{bmatrix} 2 & 0 \\ 1 & 3 \end{bmatrix} \]
\[ \text{Eigenvalues: } \lambda_1 = 2, \quad \lambda_2 = 3 \]
\[ \text{Untuk } \lambda_1 = 2: (A - 2I)v = 0 \quad \Rightarrow \quad \begin{bmatrix} 0 & 0 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \]
\[ y = -x \quad \Rightarrow \quad v_1 = \begin{bmatrix} 1 \\ -1 \end{bmatrix} \]
\[ \text{Untuk } \lambda_2 = 3: (A - 3I)v = 0 \quad \Rightarrow \quad \begin{bmatrix} -1 & 0 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \]
\[ x = 0 \quad \Rightarrow \quad v_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \]
Syntax R untuk eigen value dan eigen vector:
eigX = eigen(X); eigX## eigen() decomposition
## $values
## [1] 1.611684e+01 -1.116844e+00 -1.303678e-15
##
## $vectors
## [,1] [,2] [,3]
## [1,] -0.2319707 -0.78583024 0.4082483
## [2,] -0.5253221 -0.08675134 -0.8164966
## [3,] -0.8186735 0.61232756 0.4082483eigY = eigen(Y); eigY## eigen() decomposition
## $values
## [1] 4.242429e+01 -4.242853e-01 -8.760878e-16
##
## $vectors
## [,1] [,2] [,3]
## [1,] -0.4481957 -0.73921067 0.4082483
## [2,] -0.5688793 -0.03327957 -0.8164966
## [3,] -0.6895629 0.67265152 0.4082483eigvalX = eigX$values; eigvalX## [1] 1.611684e+01 -1.116844e+00 -1.303678e-15eigvalY = eigY$values; eigvalY## [1] 4.242429e+01 -4.242853e-01 -8.760878e-16eigvecX = eigX$vectors; eigvecX## [,1] [,2] [,3]
## [1,] -0.2319707 -0.78583024 0.4082483
## [2,] -0.5253221 -0.08675134 -0.8164966
## [3,] -0.8186735 0.61232756 0.4082483eigvecY = eigY$vectors; eigvecY## [,1] [,2] [,3]
## [1,] -0.4481957 -0.73921067 0.4082483
## [2,] -0.5688793 -0.03327957 -0.8164966
## [3,] -0.6895629 0.67265152 0.40824833 DEKOMPOSISI SINGULAR VALUE (SVD)
Dekomposisi Singular Value (SVD) adalah metode untuk memecah sebuah matriks menjadi tiga bagian yang membantu kita memahami strukturnya.
Syntax SVD di R:
library(MASS)
A <- matrix(c(5,-3,6,2,-4,8,-2,5,-1,7,3,9), 4, 3, byrow=TRUE)
A## [,1] [,2] [,3]
## [1,] 5 -3 6
## [2,] 2 -4 8
## [3,] -2 5 -1
## [4,] 7 3 9svd_result <- svd(A)
singular_value <- svd_result$d ; singular_value## [1] 16.07076 7.41936 3.11187U <- svd_result$u ; U## [,1] [,2] [,3]
## [1,] -0.5046975 0.2278362 -0.3742460
## [2,] -0.5178195 0.4138180 0.7413297
## [3,] 0.1646416 -0.6063789 0.5337354
## [4,] -0.6708477 -0.6396483 -0.1596770V <- svd_result$v ; V## [,1] [,2] [,3]
## [1,] -0.5341591 -0.17494276 -0.8270847
## [2,] 0.1490928 -0.98251336 0.1115295
## [3,] -0.8321330 -0.06373793 0.55090114 MATRIKS JARAK
Matriks jarak adalah sebuah matriks yang menunjukkan seberapa jauh atau berbeda tiap observasi (baris) satu sama lain dalam sebuah dataset.
Syntax untuk matriks jarak di R:
set.seed(321)
ss <- sample(1:50, 15)
df <- USArrests[ss, ]
df.scaled <- scale(df); df.scaled## Murder Assault UrbanPop Rape
## Wyoming -0.3721741 -0.02296746 -0.3418930 -0.62039386
## Illinois 0.4221896 1.02244775 1.2520675 0.62633064
## Mississippi 1.6799322 1.14124493 -1.4507350 -0.39776448
## Kansas -0.5486994 -0.56943449 0.0739228 -0.26418686
## New York 0.5766492 1.08184634 1.4599754 0.93801176
## Kentucky 0.2677300 -0.64071280 -0.8963140 -0.51650015
## Oklahoma -0.4163054 -0.14176464 0.2125281 0.03265231
## Hawaii -0.7031590 -1.38913505 1.2520675 0.06233622
## Missouri 0.1132704 0.17898775 0.3511333 1.24969289
## New Mexico 0.6428462 1.45011760 0.3511333 1.82852926
## Louisiana 1.5254725 1.02244775 0.0739228 0.35917539
## South Dakota -1.0341439 -0.91394632 -1.3814324 -1.03596869
## Iowa -1.3871944 -1.27033787 -0.5498008 -1.25859806
## North Dakota -1.6961136 -1.40101477 -1.4507350 -1.85227639
## Texas 0.9296998 0.45222127 1.0441596 0.84896001
## attr(,"scaled:center")
## Murder Assault UrbanPop Rape
## 8.486667 162.933333 64.933333 19.780000
## attr(,"scaled:scale")
## Murder Assault UrbanPop Rape
## 4.531929 84.177081 14.429467 6.7376554.1 Jarak Euclidean
Euclidean → Jarak lurus (garis terpendek) antara dua titik di ruang 𝑝-dimensi [jarak lurus standar].
Contoh aplikasi: - menghitung jarak garis lurus antar dua koordinat (GPS) - Clustering (K-Means, Hierarchical) → objek yang jaraknya dekat digabungkan
Syntax Jarak Euclidean di R:
library(factoextra)## Loading required package: ggplot2## Warning: package 'ggplot2' was built under R version 4.4.3## Welcome! Want to learn more? See two factoextra-related books at https://goo.gl/ve3WBadist.eucl <- dist(df.scaled, method = "euclidean"); dist.eucl## Wyoming Illinois Mississippi Kansas New York Kentucky
## Illinois 2.4122476
## Mississippi 2.6164146 3.1543527
## Kansas 0.7934567 2.3786048 3.1993198
## New York 2.7921742 0.4095812 3.3878156 2.7128511
## Kentucky 1.0532156 2.9515362 2.3433244 1.2948587 3.2757206
## Oklahoma 0.8659748 1.8685718 2.9986711 0.5547563 2.2043102 1.4993175
## Hawaii 2.2322175 2.7203365 4.4270510 1.4800030 2.9246694 2.5403456
## Missouri 2.0625111 1.4167282 3.0563398 1.8349434 1.5351057 2.3176129
## New Mexico 3.1109091 1.5775154 3.0617092 3.1551035 1.4705638 3.4011133
## Louisiana 2.4137967 1.6360410 1.7133330 2.6879097 1.7776353 2.4609320
## South Dakota 1.5765126 3.9457686 3.4644086 1.7515852 4.3067435 1.5082173
## Iowa 1.7426214 3.9154083 4.0958166 1.6038155 4.2724405 1.9508929
## North Dakota 2.5296038 4.8794481 4.4694938 2.6181473 5.2524274 2.5546862
## Texas 2.4496576 0.8218968 2.9692463 2.3259192 0.8377979 2.6949264
## Oklahoma Hawaii Missouri New Mexico Louisiana South Dakota
## Illinois
## Mississippi
## Kansas
## New York
## Kentucky
## Oklahoma
## Hawaii 1.6491638
## Missouri 1.3724911 2.3123720
## New Mexico 2.6268378 3.7154012 1.4937447
## Louisiana 2.2916633 3.5012381 1.8909275 1.7882330
## South Dakota 2.1588538 2.9115203 3.2767510 4.4281177 3.7902169
## Iowa 2.1130016 2.3395756 3.3845451 4.6758935 4.0922753 0.9964108
## North Dakota 3.0891779 3.4578871 4.3173165 5.5131433 4.8442635 1.1604313
## Texas 1.8768374 2.5920693 1.1756214 1.5867966 1.3643137 3.8935265
## Iowa North Dakota
## Illinois
## Mississippi
## Kansas
## New York
## Kentucky
## Oklahoma
## Hawaii
## Missouri
## New Mexico
## Louisiana
## South Dakota
## Iowa
## North Dakota 1.1298867
## Texas 3.9137858 4.8837032fviz_dist(dist.eucl)
4.2 Jarak Chebyshev
Chebyshev → jarak ditentukan oleh selisih terbesar. Jarak maksimum di antara perbedaan koordinat. Fokus pada dimensi dengan selisih terbesar.
Contoh aplikasi: - jarak langkah raja antara dua posisi = jarak Chebyshev. - Berguna di quality control multivariat, misalnya mengecek dimensi produk (lebar, panjang, tinggi) → fokus pada dimensi terburuk.
Syntax Jarak Chebyshev di R:
dist.cheb <- dist(df.scaled, method = "maximum"); dist.cheb## Wyoming Illinois Mississippi Kansas New York Kentucky
## Illinois 1.5939604
## Mississippi 2.0521063 2.7028025
## Kansas 0.5464670 1.5918822 2.2286315
## New York 1.8018683 0.3116811 2.9107104 1.6512808
## Kentucky 0.6399041 2.1483815 1.7819577 0.9702368 2.3562894
## Oklahoma 0.6530462 1.1642124 2.0962376 0.4276699 1.2474473 1.1088421
## Hawaii 1.5939604 2.4115828 2.7028025 1.1781447 2.4709814 2.1483815
## Missouri 1.8700867 0.9009342 1.8018683 1.5138797 1.1088421 1.7661930
## New Mexico 2.4489231 1.2021986 2.2262937 2.0927161 1.1088421 2.3450294
## Louisiana 1.8976467 1.1781447 1.5246578 2.0741719 1.3860526 1.6631605
## South Dakota 1.0395394 2.6334999 2.7140760 1.4553552 2.8414078 1.3018739
## Iowa 1.2473704 2.2927856 3.0671266 0.9944112 2.3521842 1.6549244
## North Dakota 1.3780473 2.7028025 3.3760458 1.5880895 2.9107104 1.9638436
## Texas 1.4693539 0.5702265 2.4948946 1.4783991 0.6296251 1.9404736
## Oklahoma Hawaii Missouri New Mexico Louisiana South Dakota
## Illinois
## Mississippi
## Kansas
## New York
## Kentucky
## Oklahoma
## Hawaii 1.2473704
## Missouri 1.2170406 1.5681228
## New Mexico 1.7958770 2.8392526 1.2711298
## Louisiana 1.9417780 2.4115828 1.4122022 1.4693539
## South Dakota 1.5939604 2.6334999 2.2856616 2.8644979 2.5596164
## Iowa 1.2912504 1.8018683 2.5082909 3.0871273 2.9126670 0.8316315
## North Dakota 1.8849287 2.7028025 3.1019693 3.6808057 3.2215862 0.8163077
## Texas 1.3460052 1.8413563 0.8164294 0.9978963 0.9702368 2.4255920
## Iowa North Dakota
## Illinois
## Mississippi
## Kansas
## New York
## Kentucky
## Oklahoma
## Hawaii
## Missouri
## New Mexico
## Louisiana
## South Dakota
## Iowa
## North Dakota 0.9009342
## Texas 2.3168942 2.7012364fviz_dist(dist.cheb)
4.3 Jarak Manhattan
Manhattan → Jumlah perbedaan absolut antar koordinat, seperti berjalan di jalan kota berbentuk grid [jarak berbasis grid (jumlah selisih)].
Contoh aplikasi: - menghitung jarak dalam gudang/grid jalan yang tidak memungkinkan jalur diagonal. - menghitung jarak antar dokumen berdasarkan frekuensi kata (NLP).
Syntax Jarak Manhattan di R:
dist.man <- dist(df.scaled, method = "manhattan"); dist.man## Wyoming Illinois Mississippi Kansas New York Kentucky
## Illinois 4.6804639
## Mississippi 4.5477901 5.1034373
## Kansas 1.4950151 4.6314334 5.5975464
## New York 5.4139111 0.7334472 5.4091682 5.3648806
## Kentucky 1.9159642 5.1088324 3.8673166 2.1102578 5.8422796
## Oklahoma 1.3703957 3.6359252 5.4729270 0.9955082 4.3693724 2.8409781
## Hawaii 3.9738430 4.1009258 8.0763743 2.4788279 4.8343730 4.4465291
## Missouri 3.2505127 2.6766756 5.9782446 3.2014823 2.7867606 3.9878005
## New Mexico 5.6300548 2.7514592 5.3741207 5.5810243 2.4338278 6.0584233
## Louisiana 4.3384469 2.5485829 2.5548545 4.2894164 2.9731109 4.7668154
## South Dakota 3.0080629 7.6885267 5.4767741 3.0570933 8.4219740 2.5796943
## Iowa 3.1085028 7.7889667 7.2404771 3.1575333 8.5224139 3.3731605
## North Dakota 5.0427114 9.7231753 7.3728174 5.0917419 10.4566225 4.6143429
## Texas 4.6324690 1.5082739 5.1808752 4.5834386 1.4875431 5.0608376
## Oklahoma Hawaii Missouri New Mexico Louisiana
## Illinois
## Mississippi
## Kansas
## New York
## Kentucky
## Oklahoma
## Hawaii 2.6034473
## Missouri 2.2059740 4.4728430
## New Mexico 4.5855161 6.8523850 2.3795420
## Louisiana 3.5711187 6.1151982 3.4233902 3.0568606
## South Dakota 4.0526016 4.5379784 6.2585756 8.6381176 7.3465098
## Iowa 4.1530415 3.9256352 6.3590155 8.7385576 7.4469497
## North Dakota 6.0872501 5.6222495 8.2932241 10.6727662 9.3811583
## Texas 3.5879303 4.4687467 2.1834220 2.9573454 2.6260207
## South Dakota Iowa North Dakota
## Illinois
## Mississippi
## Kansas
## New York
## Kentucky
## Oklahoma
## Hawaii
## Missouri
## New Mexico
## Louisiana
## South Dakota
## Iowa 1.7637030
## North Dakota 2.0346485 1.9342086
## Texas 7.6405319 7.7409718 9.6751804fviz_dist(dist.man)
4.4 Jarak Mahalanobis
Mahalanobis → Jarak antar titik yang mempertimbangkan skala (varians) dan korelasi antar variabel.
Contoh aplikasi: - misalnya mendeteksi transaksi keuangan yang tidak wajar - memisahkan kelompok dengan varians dan korelasi berbeda (Analisis Diskriminan)
Syntax Jarak Mahalanobis di R:
center <- colMeans(df.scaled)
cov_matrix <- cov(df.scaled)
dist.mah <- mahalanobis(df.scaled, center, cov_matrix)
dist.mah## Wyoming Illinois Mississippi Kansas New York Kentucky
## 1.4833812 4.3099818 7.1777961 0.4220713 3.7281363 3.8406053
## Oklahoma Hawaii Missouri New Mexico Louisiana South Dakota
## 0.4255505 7.0025193 4.7697204 7.7270275 3.1431292 2.7059402
## Iowa North Dakota Texas
## 2.2743931 4.4972638 2.49248395 VEKTOR RATA-RATA
Vektor rata-rata adalah vektor yang berisi nilai rata-rata dari setiap peubah (variabel) dalam suatu data multivariat.
5.1 Matriks rata-rata
Matriks rata-rata adalah matriks yang dibentuk dengan mengulang vektor rata-rata pada setiap baris sebanyak jumlah observasi, sehingga seluruh baris dalam matriks tersebut identik dengan vektor rata-rata.
Syntax matriks rata-rata di R:
#input data kadal
BB = c(6.2,11.5,8.7,10.1,7.8,6.9,12.0,3.1,14.8,9.4)
PM = c(61,73,68,70,64,60,76,49,84,71)
RTB = c(115,138,127,123,131,120,143,95,160,128)
lizard = as.matrix(cbind(BB,PM,RTB)); lizard## BB PM RTB
## [1,] 6.2 61 115
## [2,] 11.5 73 138
## [3,] 8.7 68 127
## [4,] 10.1 70 123
## [5,] 7.8 64 131
## [6,] 6.9 60 120
## [7,] 12.0 76 143
## [8,] 3.1 49 95
## [9,] 14.8 84 160
## [10,] 9.4 71 128#matriks rata-rata
vecMeans = as.matrix(colMeans(lizard)); vecMeans## [,1]
## BB 9.05
## PM 67.60
## RTB 128.00vecRata = matrix(c(mean(BB), mean(PM), mean(RTB)), nrow=3, ncol=1); vecRata## [,1]
## [1,] 9.05
## [2,] 67.60
## [3,] 128.005.2 Matriks kovarians
Matriks kovarians adalah matriks persegi yang menunjukkan seberapa banyak dua atau lebih variabel acak bervariasi bersama-sama; elemen diagonal utama matriks berisi variansi dari masing-masing variabel, sedangkan elemen di luar diagonal utama menunjukkan kovarians antar pasangan variabel.
Contoh: \[ \Sigma = \begin{bmatrix} \mathrm{Var}(X_1) & \mathrm{Cov}(X_1, X_2) & \cdots & \mathrm{Cov}(X_1, X_p) \\ \mathrm{Cov}(X_2, X_1) & \mathrm{Var}(X_2) & \cdots & \mathrm{Cov}(X_2, X_p) \\ \vdots & \vdots & \ddots & \vdots \\ \mathrm{Cov}(X_p, X_1) & \mathrm{Cov}(X_p, X_2) & \cdots & \mathrm{Var}(X_p) \end{bmatrix} \]
Syntax matriks kovarians di R:
#input data kadal
BB = c(6.2,11.5,8.7,10.1,7.8,6.9,12.0,3.1,14.8,9.4)
PM = c(61,73,68,70,64,60,76,49,84,71)
RTB = c(115,138,127,123,131,120,143,95,160,128)
lizard = as.matrix(cbind(BB,PM,RTB)); lizard## BB PM RTB
## [1,] 6.2 61 115
## [2,] 11.5 73 138
## [3,] 8.7 68 127
## [4,] 10.1 70 123
## [5,] 7.8 64 131
## [6,] 6.9 60 120
## [7,] 12.0 76 143
## [8,] 3.1 49 95
## [9,] 14.8 84 160
## [10,] 9.4 71 128#matriks kovarians
varkov = cov(lizard); varkov## BB PM RTB
## BB 10.98056 31.80000 54.96667
## PM 31.80000 94.04444 160.22222
## RTB 54.96667 160.22222 300.666675.3 Matriks korelasi
Matriks korelasi yaitu sebuah matriks dengan elemen-elemen matriks yang merupakan koefisien korelasi dengan nilai terletak pada interval [−1,1] dan khusus elemen diagonal matriks bernilai satu.
Contoh: \[ R = \begin{bmatrix} 1 & r_{12} & r_{13} & \cdots & r_{1p} \\ r_{21} & 1 & r_{23} & \cdots & r_{2p} \\ r_{31} & r_{32} & 1 & \cdots & r_{3p} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ r_{p1} & r_{p2} & r_{p3} & \cdots & 1 \end{bmatrix} \]
Syntax matriks korelasi di R:
#input data kadal
BB = c(6.2,11.5,8.7,10.1,7.8,6.9,12.0,3.1,14.8,9.4)
PM = c(61,73,68,70,64,60,76,49,84,71)
RTB = c(115,138,127,123,131,120,143,95,160,128)
lizard = as.matrix(cbind(BB,PM,RTB)); lizard## BB PM RTB
## [1,] 6.2 61 115
## [2,] 11.5 73 138
## [3,] 8.7 68 127
## [4,] 10.1 70 123
## [5,] 7.8 64 131
## [6,] 6.9 60 120
## [7,] 12.0 76 143
## [8,] 3.1 49 95
## [9,] 14.8 84 160
## [10,] 9.4 71 128#matriks korelasi
korel = cor(lizard); korel## BB PM RTB
## BB 1.0000000 0.9895743 0.9566313
## PM 0.9895743 1.0000000 0.9528259
## RTB 0.9566313 0.9528259 1.00000005.4 Matriks standardisasi
Matriks standarisasi adalah matriks yang diperoleh dengan mentransformasi data asli sehingga setiap variabel memiliki rata-rata nol dan simpangan baku satu. Proses standarisasi dilakukan dengan mengurangkan rata-rata variabel dari setiap observasi, kemudian membaginya dengan simpangan baku variabel tersebut.
Syntax matriks standarisasi di R:
#input data kadal
BB = c(6.2,11.5,8.7,10.1,7.8,6.9,12.0,3.1,14.8,9.4)
PM = c(61,73,68,70,64,60,76,49,84,71)
RTB = c(115,138,127,123,131,120,143,95,160,128)
lizard = as.matrix(cbind(BB,PM,RTB)); lizard## BB PM RTB
## [1,] 6.2 61 115
## [2,] 11.5 73 138
## [3,] 8.7 68 127
## [4,] 10.1 70 123
## [5,] 7.8 64 131
## [6,] 6.9 60 120
## [7,] 12.0 76 143
## [8,] 3.1 49 95
## [9,] 14.8 84 160
## [10,] 9.4 71 128#matriks standarisasi
n = nrow(lizard);n## [1] 10u = matrix(1,n,1); u## [,1]
## [1,] 1
## [2,] 1
## [3,] 1
## [4,] 1
## [5,] 1
## [6,] 1
## [7,] 1
## [8,] 1
## [9,] 1
## [10,] 1xbar = cbind((1/n)*t(u)%*%lizard); xbar## BB PM RTB
## [1,] 9.05 67.6 128D = lizard - u %*% xbar; D## BB PM RTB
## [1,] -2.85 -6.6 -13
## [2,] 2.45 5.4 10
## [3,] -0.35 0.4 -1
## [4,] 1.05 2.4 -5
## [5,] -1.25 -3.6 3
## [6,] -2.15 -7.6 -8
## [7,] 2.95 8.4 15
## [8,] -5.95 -18.6 -33
## [9,] 5.75 16.4 32
## [10,] 0.35 3.4 0S = (1/(n-1))*t(D)%*%D; S## BB PM RTB
## BB 10.98056 31.80000 54.96667
## PM 31.80000 94.04444 160.22222
## RTB 54.96667 160.22222 300.66667Ds = diag(sqrt(diag(S))); Ds## [,1] [,2] [,3]
## [1,] 3.313692 0.000000 0.00000
## [2,] 0.000000 9.697651 0.00000
## [3,] 0.000000 0.000000 17.33974R = solve(Ds) %*% S %*% solve(Ds); R## [,1] [,2] [,3]
## [1,] 1.0000000 0.9895743 0.9566313
## [2,] 0.9895743 1.0000000 0.9528259
## [3,] 0.9566313 0.9528259 1.0000000