Analisis Data Multivariat 1
Stefanie Aurelia Clarissa H.
2025-09-08
Perkenalan Matriks
Analisis Data Multivariat adalah analisis data yang sangat identik dengan penggunaan matriks. Maka dari itu, diperlukan pemahaman ulang serta penggunaan syntax matriks pada R.
Membuat Matriks
A <- matrix(c(1,4,6,13,65,55,8,1,4), nrow=3, byrow=TRUE)
B <- matrix(c(4,1,8,55,65,13,6,4,1), nrow=3, byrow=TRUE)
A; B## [,1] [,2] [,3]
## [1,] 1 4 6
## [2,] 13 65 55
## [3,] 8 1 4## [,1] [,2] [,3]
## [1,] 4 1 8
## [2,] 55 65 13
## [3,] 6 4 1library(factoextra) ## Warning: package 'factoextra' was built under R version 4.4.3## Loading required package: ggplot2## Welcome! Want to learn more? See two factoextra-related books at https://goo.gl/ve3WBaA + B # Penjumlahan## [,1] [,2] [,3]
## [1,] 5 5 14
## [2,] 68 130 68
## [3,] 14 5 5A %*% B # Perkalian Matriks## [,1] [,2] [,3]
## [1,] 260 285 66
## [2,] 3957 4458 1004
## [3,] 111 89 81t(A) # Transpose## [,1] [,2] [,3]
## [1,] 1 13 8
## [2,] 4 65 1
## [3,] 6 55 4det(A) # Determinan## [1] -1285library(Matrix)## Warning: package 'Matrix' was built under R version 4.4.3rankMatrix(A)## [1] 3
## attr(,"method")
## [1] "tolNorm2"
## attr(,"useGrad")
## [1] FALSE
## attr(,"tol")
## [1] 6.661338e-16A %*% A## [,1] [,2] [,3]
## [1,] 101 270 250
## [2,] 1298 4332 3873
## [3,] 53 101 119# Memanggil Komponen Matriks
A## [,1] [,2] [,3]
## [1,] 1 4 6
## [2,] 13 65 55
## [3,] 8 1 4A[,2] # Kolom 2## [1] 4 65 1A[3,] # Baris 3## [1] 8 1 4A[3,2] # Sel(3, 2)## [1] 1A[c(1,3),2] # Sel(1,2) dan sel(3,2)## [1] 4 1A[,1:3] # kolom(1,2,3)## [,1] [,2] [,3]
## [1,] 1 4 6
## [2,] 13 65 55
## [3,] 8 1 4A[2:3,] # baris(2,3)## [,1] [,2] [,3]
## [1,] 13 65 55
## [2,] 8 1 4Q <- matrix(c(0, -1, 1, 0), nrow=2, byrow=TRUE)
t(Q) %*% Q # Harus = matriks identitas## [,1] [,2]
## [1,] 1 0
## [2,] 0 1inv_Q = solve(Q); inv_Q # invers## [,1] [,2]
## [1,] 0 1
## [2,] -1 0I <- diag(3)
I## [,1] [,2] [,3]
## [1,] 1 0 0
## [2,] 0 1 0
## [3,] 0 0 1#------------------------------------------Eigen Value
# Eigen Value menunjukkan jumlah variasi (informasi) yang dapat dijelaskan oleh satu komponen
# Eigen Vector menunjukkan jumlah variasi (informasi) yang dijelaskan suatu komponen lainnya
# Eigen Value dan eigen vector ini sangat berguna dalam analisis data multivariat misalnya Principal Komponen Analysis, Factor Analysis, dll.
M <- matrix(c(2, -1, -1, 2), nrow=2)
eigen(M)$values # semua positif → definit positif## [1] 3 1N <- -M
eigen(N)$values # semua negatif → definit negatif## [1] -1 -3eig_A <- eigen(A)
eig_A$values## [1] 67.122514 6.044618 -3.167131eig_A$vectors## [,1] [,2] [,3]
## [1,] -0.06251160 -0.2591413 0.5636637
## [2,] -0.99776211 0.6887849 0.4509012
## [3,] -0.02372933 -0.6770680 -0.6920776set.seed(123)
v <- rnorm(5) # Vektor acak
M_random <- matrix(runif(9, min=0, max=10), nrow=3)
v; M_random## [1] -0.56047565 -0.23017749 1.55870831 0.07050839 0.12928774## [,1] [,2] [,3]
## [1,] 9.568333 5.726334 2.4608773
## [2,] 4.533342 1.029247 0.4205953
## [3,] 6.775706 8.998250 3.2792072# Dekomposisi Singular Value (SVD)
## SVD memecah sebuah matriks p 4mCn menjadi tiga matriks
library(MASS)## Warning: package 'MASS' was built under R version 4.4.3A <- 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 0.5509011# MATRIKS JARAK #
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 6.737655## 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
dist.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)
## 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.
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)
## 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).
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)
## 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)
library(StatMatch)## Warning: package 'StatMatch' was built under R version 4.4.3## Loading required package: proxy## Warning: package 'proxy' was built under R version 4.4.3##
## Attaching package: 'proxy'## The following object is masked from 'package:Matrix':
##
## as.matrix## The following objects are masked from 'package:stats':
##
## as.dist, dist## The following object is masked from 'package:base':
##
## as.matrix## Loading required package: survey## Warning: package 'survey' was built under R version 4.4.3## Loading required package: grid## Loading required package: survival##
## Attaching package: 'survey'## The following object is masked from 'package:graphics':
##
## dotchart## Loading required package: lpSolve## Warning: package 'lpSolve' was built under R version 4.4.2## Loading required package: dplyr##
## Attaching package: 'dplyr'## The following object is masked from 'package:MASS':
##
## select## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, uniondist.mah <- mahalanobis.dist(df.scaled); dist.mah## Wyoming Illinois Mississippi Kansas New York Kentucky
## Wyoming 0.000000 1.7186109 2.820779 1.4195095 1.8695558 2.867847
## Illinois 1.718611 0.0000000 3.658323 2.2905255 0.4722069 3.878642
## Mississippi 2.820779 3.6583235 0.000000 3.2139075 3.6566922 2.544477
## Kansas 1.419510 2.2905255 3.213907 0.0000000 2.1522535 2.048031
## New York 1.869556 0.4722069 3.656692 2.1522535 0.0000000 3.698342
## Kentucky 2.867847 3.8786421 2.544477 2.0480310 3.6983422 0.000000
## Oklahoma 1.146496 1.8980286 3.237573 0.6499978 1.7772007 2.505941
## Hawaii 3.466671 3.6449604 4.722203 2.2108491 3.3748818 2.753554
## Missouri 3.198071 3.6796400 3.956918 2.2592572 3.3618939 2.642756
## New Mexico 3.281318 3.5101406 4.057258 3.1016653 3.2869855 3.870023
## Louisiana 2.284940 2.5550539 1.688058 2.2700723 2.4136664 2.119635
## South Dakota 1.826205 3.3564158 3.087365 1.6274307 3.3404110 2.261154
## Iowa 1.327907 2.6329606 3.559587 1.1128197 2.6839965 2.621704
## North Dakota 1.582582 3.1919907 3.553572 1.9466491 3.3317039 3.040465
## Texas 2.540604 2.4769381 3.093919 1.7462066 2.1399545 2.108949
## Oklahoma Hawaii Missouri New Mexico Louisiana South Dakota
## Wyoming 1.1464956 3.466671 3.198071 3.281318 2.284940 1.826205
## Illinois 1.8980286 3.644960 3.679640 3.510141 2.555054 3.356416
## Mississippi 3.2375727 4.722203 3.956918 4.057258 1.688058 3.087365
## Kansas 0.6499978 2.210849 2.259257 3.101665 2.270072 1.627431
## New York 1.7772007 3.374882 3.361894 3.286985 2.413666 3.340411
## Kentucky 2.5059414 2.753554 2.642756 3.870023 2.119635 2.261154
## Oklahoma 0.0000000 2.705865 2.203038 2.660216 2.350208 1.672866
## Hawaii 2.7058650 0.000000 3.193764 4.645567 3.383255 3.551072
## Missouri 2.2030382 3.193764 0.000000 1.836797 3.256319 2.505784
## New Mexico 2.6602159 4.645567 1.836797 0.000000 3.676879 3.026024
## Louisiana 2.3502077 3.383255 3.256319 3.676879 0.000000 3.021642
## South Dakota 1.6728664 3.551072 2.505784 3.026024 3.021642 0.000000
## Iowa 1.3299426 2.790197 3.145245 3.792086 2.954252 1.518854
## North Dakota 1.9813596 3.780966 3.590548 3.950259 3.434074 1.304743
## Texas 1.9635201 2.082005 2.576037 3.501666 1.527269 3.090805
## Iowa North Dakota Texas
## Wyoming 1.327907 1.582582 2.540604
## Illinois 2.632961 3.191991 2.476938
## Mississippi 3.559587 3.553572 3.093919
## Kansas 1.112820 1.946649 1.746207
## New York 2.683996 3.331704 2.139954
## Kentucky 2.621704 3.040465 2.108949
## Oklahoma 1.329943 1.981360 1.963520
## Hawaii 2.790197 3.780966 2.082005
## Missouri 3.145245 3.590548 2.576037
## New Mexico 3.792086 3.950259 3.501666
## Louisiana 2.954252 3.434074 1.527269
## South Dakota 1.518854 1.304743 3.090805
## Iowa 0.000000 1.045923 2.734770
## North Dakota 1.045923 0.000000 3.563193
## Texas 2.734770 3.563193 0.000000dist.mah_matrix <- as.matrix(dist.mah);dist.mah_matrix## Wyoming Illinois Mississippi Kansas New York Kentucky
## Wyoming 0.000000 1.7186109 2.820779 1.4195095 1.8695558 2.867847
## Illinois 1.718611 0.0000000 3.658323 2.2905255 0.4722069 3.878642
## Mississippi 2.820779 3.6583235 0.000000 3.2139075 3.6566922 2.544477
## Kansas 1.419510 2.2905255 3.213907 0.0000000 2.1522535 2.048031
## New York 1.869556 0.4722069 3.656692 2.1522535 0.0000000 3.698342
## Kentucky 2.867847 3.8786421 2.544477 2.0480310 3.6983422 0.000000
## Oklahoma 1.146496 1.8980286 3.237573 0.6499978 1.7772007 2.505941
## Hawaii 3.466671 3.6449604 4.722203 2.2108491 3.3748818 2.753554
## Missouri 3.198071 3.6796400 3.956918 2.2592572 3.3618939 2.642756
## New Mexico 3.281318 3.5101406 4.057258 3.1016653 3.2869855 3.870023
## Louisiana 2.284940 2.5550539 1.688058 2.2700723 2.4136664 2.119635
## South Dakota 1.826205 3.3564158 3.087365 1.6274307 3.3404110 2.261154
## Iowa 1.327907 2.6329606 3.559587 1.1128197 2.6839965 2.621704
## North Dakota 1.582582 3.1919907 3.553572 1.9466491 3.3317039 3.040465
## Texas 2.540604 2.4769381 3.093919 1.7462066 2.1399545 2.108949
## Oklahoma Hawaii Missouri New Mexico Louisiana South Dakota
## Wyoming 1.1464956 3.466671 3.198071 3.281318 2.284940 1.826205
## Illinois 1.8980286 3.644960 3.679640 3.510141 2.555054 3.356416
## Mississippi 3.2375727 4.722203 3.956918 4.057258 1.688058 3.087365
## Kansas 0.6499978 2.210849 2.259257 3.101665 2.270072 1.627431
## New York 1.7772007 3.374882 3.361894 3.286985 2.413666 3.340411
## Kentucky 2.5059414 2.753554 2.642756 3.870023 2.119635 2.261154
## Oklahoma 0.0000000 2.705865 2.203038 2.660216 2.350208 1.672866
## Hawaii 2.7058650 0.000000 3.193764 4.645567 3.383255 3.551072
## Missouri 2.2030382 3.193764 0.000000 1.836797 3.256319 2.505784
## New Mexico 2.6602159 4.645567 1.836797 0.000000 3.676879 3.026024
## Louisiana 2.3502077 3.383255 3.256319 3.676879 0.000000 3.021642
## South Dakota 1.6728664 3.551072 2.505784 3.026024 3.021642 0.000000
## Iowa 1.3299426 2.790197 3.145245 3.792086 2.954252 1.518854
## North Dakota 1.9813596 3.780966 3.590548 3.950259 3.434074 1.304743
## Texas 1.9635201 2.082005 2.576037 3.501666 1.527269 3.090805
## Iowa North Dakota Texas
## Wyoming 1.327907 1.582582 2.540604
## Illinois 2.632961 3.191991 2.476938
## Mississippi 3.559587 3.553572 3.093919
## Kansas 1.112820 1.946649 1.746207
## New York 2.683996 3.331704 2.139954
## Kentucky 2.621704 3.040465 2.108949
## Oklahoma 1.329943 1.981360 1.963520
## Hawaii 2.790197 3.780966 2.082005
## Missouri 3.145245 3.590548 2.576037
## New Mexico 3.792086 3.950259 3.501666
## Louisiana 2.954252 3.434074 1.527269
## South Dakota 1.518854 1.304743 3.090805
## Iowa 0.000000 1.045923 2.734770
## North Dakota 1.045923 0.000000 3.563193
## Texas 2.734770 3.563193 0.000000# Jarak Minkowski
# Jarak Minkowski adalah ukuran jarak antara dua titik dalam ruang vektor yang ditentukan oleh sebuah parameter p untuk mencari jarak umum karena menjadi bentuk dasar yang mencakup berbagai jenis jarak lain
# p=1 jarak Manhattan, p=2 jarak Euclidean, p->tak hingga jarak chebyshev
set.seed(123)
# Data random (5 observasi dengan 3 variabel)
data <- matrix(runif(15, min = 1, max = 10), nrow = 5, ncol = 3)
colnames(data) <- c("X1", "X2", "X3")
print("Data random:")## [1] "Data random:"print(data)## X1 X2 X3
## [1,] 3.588198 1.410008 9.611500
## [2,] 8.094746 5.752949 5.080007
## [3,] 4.680792 9.031771 7.098136
## [4,] 8.947157 5.962915 6.153701
## [5,] 9.464206 5.109533 1.926322# Tentukan dua titik yang akan dihitung jaraknya
p1 <- data[1, ];p1## X1 X2 X3
## 3.588198 1.410008 9.611500p2 <- data[2, ];p2## X1 X2 X3
## 8.094746 5.752949 5.080007# Fungsi jarak Minkowski
minkowski_distance <- function(x, y, p) {
sum(abs(x - y)^p)^(1/p)
}
# Contoh penggunaan dengan p = 1 (Manhattan), p = 2 (Euclidean), p = 3 (Minkowski umum)
dist_p1 <- minkowski_distance(p1, p2, p = 1);dist_p1## [1] 13.38098dist_p2 <- minkowski_distance(p1, p2, p = 2);dist_p2## [1] 7.726871dist_p3 <- minkowski_distance(p1, p2, p = 3);dist_p3## [1] 6.435156dist_inf <- max(abs(p1 - p2));dist_inf## [1] 4.531493# VEKTOR RATA-RATA #
# 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,] 128.00# 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 300.66667# 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 1.0000000# Matriks Standardisasi (akar dari variansi masing-masing variabel)
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 1.0000000Note that the echo = FALSE parameter was added to the
code chunk to prevent printing of the R code that generated the
plot.