Matriks, Vektor, dan Matriks Jarak serta Penerapan Menggunakan R
David Christian
2025-09-11
MATRIKS
Matriks merupakan sekmpulan elemen-elemen yang disusun dengan baris dan kolom. Bilangan-bilangan yang berada di dalam matriks disebut elemen atau entri matriks. Matriks sendiri dinotasikan dengan huruf kapital tebal.
Cara Menginput Matriks
Pada perangkat lunak R, matriks dapat diinput berdasarkan dua cara: berdasarkan kolom dan berdasarkan barisnya. Berikut cara menginput matriks berdasarkan kolom.
X = matrix(c(6.5,8.2,7.9,
5.4,7.0,6.7,
8.1,6.9,9.2), nrow = 3, ncol = 3); X## [,1] [,2] [,3]
## [1,] 6.5 5.4 8.1
## [2,] 8.2 7.0 6.9
## [3,] 7.9 6.7 9.2Kemudian jika ingin menginput matriks berdasarkan barisnya, byrow = TRUE harus ditambahkan sebagai berikut.
Y = matrix(c(7.3,6.8,8.5,
8.9,7.6,6.1,
9.4,8.0,7.2), nrow = 3, ncol = 3, byrow = TRUE); Y## [,1] [,2] [,3]
## [1,] 7.3 6.8 8.5
## [2,] 8.9 7.6 6.1
## [3,] 9.4 8.0 7.2Operasi Dasar Matriks
Penjumlahan dan Pengurangan
Jika ada dua matriks X dan Y dengan ukuran sama, maka operasi dilakukan elemen per elemen dan operasi tersebut hanya berlaku jika matriks tersebut berdimensi atau berordo sama. Operasi ini dapat dijalankan dengan fungsi berikut.
# penjumlahan
X + Y## [,1] [,2] [,3]
## [1,] 13.8 12.2 16.6
## [2,] 17.1 14.6 13.0
## [3,] 17.3 14.7 16.4# pengurangan
X - Y## [,1] [,2] [,3]
## [1,] -0.8 -1.4 -0.4
## [2,] -0.7 -0.6 0.8
## [3,] -1.5 -1.3 2.0Y - X## [,1] [,2] [,3]
## [1,] 0.8 1.4 0.4
## [2,] 0.7 0.6 -0.8
## [3,] 1.5 1.3 -2.0Perkalian
Perkalian matriks berbeda dengan perkalian elemen. Perkalian matriks menggunakan kombinasi baris dikali kolom sebagai berikut.
# perkalian
X %*% Y## [,1] [,2] [,3]
## [1,] 171.65 150.04 146.51
## [2,] 187.02 164.16 162.08
## [3,] 203.78 178.24 174.26Y %*% X## [,1] [,2] [,3]
## [1,] 170.36 143.97 184.25
## [2,] 168.36 142.13 180.65
## [3,] 183.58 155.00 197.58Selain itu, jika ingin melakukan perkalian skalar dengan suatu matriks atau bahkan hanya ingin mengalikan elemen-elemennya, tanpa baris dikali kolom, dilakukan sebagai berikut.
# perkalian dengan skalar
2*X## [,1] [,2] [,3]
## [1,] 13.0 10.8 16.2
## [2,] 16.4 14.0 13.8
## [3,] 15.8 13.4 18.4# perkalian elemen
X*Y## [,1] [,2] [,3]
## [1,] 47.45 36.72 68.85
## [2,] 72.98 53.20 42.09
## [3,] 74.26 53.60 66.24Transpose Matriks
Transpose matriks bertujuan untuk mengubah baris menjadi kolom.
# transpose
transX = t(X); transX## [,1] [,2] [,3]
## [1,] 6.5 8.2 7.9
## [2,] 5.4 7.0 6.7
## [3,] 8.1 6.9 9.2transY = t(Y); transY## [,1] [,2] [,3]
## [1,] 7.3 8.9 9.4
## [2,] 6.8 7.6 8.0
## [3,] 8.5 6.1 7.2Invers
Invers X^-1 memenuhi: \[ XX^-1 = X^-1X = I \] Matriks hanya punya invers jika determinannya ≠ 0. Berikut fungsi invers pada R.
# invers
inv_X = solve(X); inv_X## [,1] [,2] [,3]
## [1,] 8.3848639 2.118136 -8.9709275
## [2,] -9.6585141 -1.933549 9.9538533
## [3,] -0.1661283 -0.410706 0.5629903inv_Y = solve(Y); inv_Y## [,1] [,2] [,3]
## [1,] -1.27147766 -4.089347 4.965636
## [2,] 1.44759450 5.871993 -6.683849
## [3,] 0.05154639 -1.185567 1.082474Determinan
Determinan adalah bilangan skalar yang merepresentasikan faktor skala transformasi linear. Jika determinan = 0, matriks menjadi singular (tidak punya invers). Untuk suatu matriks X berordo 2x2 dengan elemen-elemen [a b c d], determinannya dirumuskan sebagai: \[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \to det(A) =ad-bc \] Namun, untuk matriks berordo lebih besar dari 2x2 akan lebih kompleks sehingga digunakan fungsi berikut.
det(X)## [1] 2.167det(Y)## [1] -4.656Memanggil Komponen Matriks
Pada R, komponen-komponen pada suatu matriks dapat di dipanggil sebagai berikut
X # Memanggil keseluruhan matriks## [,1] [,2] [,3]
## [1,] 6.5 5.4 8.1
## [2,] 8.2 7.0 6.9
## [3,] 7.9 6.7 9.2X[,2] # Kolom 2## [1] 5.4 7.0 6.7X[3,] # Baris 3## [1] 7.9 6.7 9.2X[3,2] # Sel(3, 2)## [1] 6.7X[c(1,3),2] # Sel(1,2) dan sel(3,2)## [1] 5.4 6.7X[,1:3] # kolom(1,2,3)## [,1] [,2] [,3]
## [1,] 6.5 5.4 8.1
## [2,] 8.2 7.0 6.9
## [3,] 7.9 6.7 9.2Nilai dan Vektor Eigen
Eigen Value menunjukkan jumlah variasi (informasi) yang dapat dijelaskan oleh satu komponen. Nilai ini dihitung dari persamaan karakteristik: \[ det(A-\lambda I) = 0 \]
Vektor Eigen menunjukkan jumlah variasi (informasi) yang dijelaskan suatu komponen lainnya. Vektor Eigen ini memenuhi persamaan: \[Av=\lambda I\]
eigX = eigen(X); eigX## eigen() decomposition
## $values
## [1] 22.0140019 0.4816027 0.2043953
##
## $vectors
## [,1] [,2] [,3]
## [1,] 0.5268942 0.5218595 0.61121520
## [2,] 0.5752916 -0.8360039 -0.78978922
## [3,] 0.6256373 0.1695881 0.05146821eigY = eigen(Y); eigY## eigen() decomposition
## $values
## [1] 23.230397 -1.286223 0.155826
##
## $vectors
## [,1] [,2] [,3]
## [1,] -0.5633011 -0.7710948 0.5522886
## [2,] -0.5584126 0.5270180 -0.8126545
## [3,] -0.6089887 0.3573023 0.1859299eigvalX = eigX$values; eigvalX## [1] 22.0140019 0.4816027 0.2043953eigvalY = eigY$values; eigvalY## [1] 23.230397 -1.286223 0.155826eigvecX = eigX$vectors; eigvecX## [,1] [,2] [,3]
## [1,] 0.5268942 0.5218595 0.61121520
## [2,] 0.5752916 -0.8360039 -0.78978922
## [3,] 0.6256373 0.1695881 0.05146821eigvecY = eigY$vectors; eigvecY## [,1] [,2] [,3]
## [1,] -0.5633011 -0.7710948 0.5522886
## [2,] -0.5584126 0.5270180 -0.8126545
## [3,] -0.6089887 0.3573023 0.1859299Singular Value Decomposition (SVD)
SVD memecah sebuah matriks 𝐴m×n menjadi tiga matriks \[A=U\Sigma V^t\]
U = matriks ortogonal m x m, V = matriks orthogonal n x n, \(\Sigma\) = matriks berukuran m x n yang elemen-elemen diagonal utamanya adalah nilai-nilai singular dari matriks A dan elemen-elemen lainnya 0
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 0.5509011Matriks Jarak
Jarak pada hal ini mengukur seberapa “mirip” atau “berbeda” dua objek. Setiap metode jarak punya interpretasi berbeda. Berikut beberapa persiapannya.
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 6.737655Jarak Euclidian
Jarak Euclidian merupakan jarak lurus (garis terpendek) antara dua titik di ruang𝑝-dimensi [jarak lurus standar]. Rumusnya adalah sebagai berikut. \[ d_{ij}=\sum_{i=1}^p|x_{ki}-x_{kj}| \] > Keterangan : > > \(x_{ki}\) = Vektor ke - k pada baris ke - i > > \(x_{kj}\)= Vektor ke - k pada baris ke - j
Berikut fungsinya pada R
install.packages("factoextra", repos = "https://cran.r-project.org")## Installing package into 'C:/Users/LENOVO/AppData/Local/R/win-library/4.4'
## (as 'lib' is unspecified)## package 'factoextra' successfully unpacked and MD5 sums checked
##
## The downloaded binary packages are in
## C:\Users\LENOVO\AppData\Local\Temp\Rtmp40LQTv\downloaded_packageslibrary(factoextra)## Warning: package 'factoextra' was built under R version 4.4.3## 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)
Jarak Chebyshev
Jarak Chebyshev merupakan jarak maksimum di antara perbedaan koordinat. Fokusnya pada dimensi dengan selisih terbesar. Berikut rumusnya: \[ d_{ij}=\max_i \, |x_{ki} - x_{kj}| \]
Keterangan :
\(x_{ki}\) = Vektor ke - k pada baris ke - i
\(x_{kj}\)= Vektor ke - k pada baris ke - j
Berikut fungsinya pada 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)
Jarak Manhattan
Jarak Manhattan merupakan jumlah perbedaan absolut antar koordinat, seperti berjalan di jalan kota berbentuk grid [jarak berbasis grid (jumlah selisih)]. Berikut rumusnya: \[ d_{ij}=\sum_{i=1}^p|x_{ki}-x_{kj}| \]
Keterangan :
\(x_{ki}\) = Vektor ke - k pada baris ke - i
\(x_{kj}\)= Vektor ke - k pada baris ke - j
Berikut perhitungannya pada 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)
Jarak Mahalanobis
Jarak Mahalanobis merupakan jarak antartitik yang mempertimbangkan skala (varians) dan korelasi antarvariabel. Dengan rumus berikut: \[ d_{ij}=\sum_{i=1}^p|x_{ki}-x_{kj}| \]
Keterangan :
\(x_{ki}\) = Vektor ke - k pada baris ke - i
\(x_{kj}\)= Vektor ke - k pada baris ke - j
Berikut perhitungannya.
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 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: Matrix## 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 0.000000Jarak 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. Rumusnya adalah sebagai berikut: Jika dua titik x dan y sebagai berikut: \[X = (x_{1},x_{2},...,x_{n}) \] \[ Dan \]
\[ Y = (y_{1},y_{2},...,y_{n}) \]
maka jarak minkowski didefinisikan sebagai berikut :
\[ D(X,Y) =(\sum_{i=1}^n|x_{i}-y_{i}|^p)^\frac{1}{p} \]
Catatan :
\(p=1\) artinya Jarak Manhattan
\(p=2\) artinya Jarak Euclidean
\(p =\infty\) artinya Jarak Chebyshev
Berikut perhitungannya pada R.
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] 4.531493Vektor Rata - rata
Misalkan ada m buah vektor berdimensi n :
\[ X = (x_{1},x_{2},...,x_{n}) \]
Maka, vektor rata - rata didefinisikan sebagai :
\[ \bar{X}=\frac{1}{m}\sum_{j=1}^mx_{j} \] Berikut contoh perhitungan pada 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 Matriks rata-rata adalah representasi nilai rata-rata dari setiap variabel dalam dataset yang disusun dalam bentuk matriks. Matriks rata-rata biasanya berupa vektor kolom yang memuat rata-rata tiap variabel.
Berikut fungsinya pada R:
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,] 128.00Matriks Kovarians
Menggambarkan seberapa besar variabel-variabel berubah bersama. Berikut rumus umumnya. \[ \Sigma = \begin{bmatrix} \text{cov}(X, X) & \text{cov}(X, Y) & \text{cov}(X, Z) \\ \text{cov}(Y, X) & \text{cov}(Y, Y) & \text{cov}(Y, Z) \\ \text{cov}(Z, X) & \text{cov}(Z, Y) & \text{cov}(Z, Z) \end{bmatrix} = \begin{bmatrix} \sigma^2_X & \sigma_{XY} & \sigma_{XZ} \\ \sigma_{YX} & \sigma^2_Y & \sigma_{YZ} \\ \sigma_{ZX} & \sigma_{ZY} & \sigma^2_Z \end{bmatrix} \]
Berikut fungsinya pada R.
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 300.66667Matriks Korelasi
Matriks korelasi adalah matriks persegi yang menunjukkan hubungan linear antar variabel dalam dataset. Berikut rumusnya: \[ R = \begin{bmatrix} 1 & r_{12} & r_{13} \\ r_{21} & 1 & r_{23} \\ r_{31} & r_{32} & 1 \end{bmatrix} \] Berikut fungsinya pada R.
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 1.0000000Matriks Standardisasi
Matriks Standardisasi secara umum adalah akar dari variansi masing-masing variabel.
Berikut contohnya pada R.
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