Cara Membuat Data Point untuk Plot (Eigen Vector)
2023-04-06
Nama : Renata Amalia Putri
NIM : 220605110074
Kelas : B
Mata Kuliah : Linear Algebra
Dosen Pengampu : Prof. Dr. Suhartono, M.kom
Jurusan : Teknik Informatika
Instansi : UIN Maulana Malik Ibrahim Malang
Agar bisa membuat titik data untuk plot, maka diperlukan menghitung eigen vectors dari matriks sigma, standar penyimpangan untuk distribusi normal bivariat. Transformasi ini menggunakan vektor eigen yang berhubungan dengan analisis komponen utama, yaitu salah satu model pembelajaran tanpa pengawasan dalam data science.
Sebagai contoh, dihasilkan titik data dari distribusi normal bivariat. Distribusi normal bivariat ditentukan oleh mean (rata-rata) dan simpangan baku, berbeda dengan distribusi normal univariat. Namun, perlu diketahui bahwa mean (rata-rata) ditentukan dengan vektor 2 dimensi dan simpangan bakunya ditentukan oleh matriks 2x2. Di bawah ini menggunakan package mvtnorm dan package matlib untuk operasi matriks.
Hal pertama yang dilakukan yaitu mengupload semua package yang dibutuhkan:
library(mvtnorm)
library(ggplot2)
library(matlib)Perlu diketahui bahwa ggplot2 adalah sebuah package yang menghasilkan statistik data atau grafik, yang dikhususkan untuk visualisasi data. Untuk membuat suatu visualisasi data maka ggplot2 didasari oleh grammar atau tata bahasa khusus. Jadi, dengan adanya ggplot2 ini memudahkan untuk merancang grafik yang menggambarkan data melalui simbol, warna, bentuk, ukuran, dan ketebalan point. Memungkinkan untuk membuat grafik yang merepresentasikan data numerik dan kategorik baik univariat maupun multivariat secara simultan.
Package mvtnorm merupakan kependekan dari Multivariate Normal and t Distributions. Jadi di package ini menyediakan fungsi untuk multivariat normal dan distribusi t. Ini terutama digunakan untuk menghitung probabilitas dan menghasilkan angka acak dari distribusi multivariat. Package ini termasuk fungsi untuk menghitung kepadatan, fungsi distribusi, fungsi kuantil, dan hasil acak untuk multivariat normal dan distribusi-t. Dan tambahan, ini juga termasuk fungsi untuk menghitung rata-rata vektor, matriks korelasi, dan komponen utama dari distribusi normal multivariat.
Kode di bawah ini, menentukan rata-rata dan standar penyimpangan dengan:
## Standar penyimpangan
sigma <- matrix(c(4,2,2,3), ncol = 2, nrow = 2)
## Rata-rata
mu <- c(1,2)Dan untuk menampilkan hasilnya cukup dengan memanggil ulang atau mengetik kode di atas yaitu sigma dan mu.
mu## [1] 1 2sigma## [,1] [,2]
## [1,] 4 2
## [2,] 2 3Untuk data ini mempunyai rata-rata dan standar penyimpangan n = 1000
n <- 1000Dan akhirnya, dapat menghasilkan data point dengan rmvnorm dan dikonversi ke data frame untuk menampilkan bingkai data/ proses data, jika data nya d maka bentuknya frame sedangkan str itu matriks.
Fungsi rmvnorm dalam bahasa R digunakan untuk menghasilkan sampel acak dari distribusi normal multivariat. Ini membutuhkan argumen seperti n (jumlah dari sampel yang dihasilkan), rata-rata (vektor rata-rata untuk tiap variabel), dan sigma (matriks kovarians dari variabel). Fungsi ini sangat berguna untuk menghasilkan data sintetik untuk simulasi studi atau pengujian metode statistik.
library MASS digunakan untuk memuat package MASS yang menyediakan fungsi dan kumpulan data untuk statistik matematika, termasuk alat untuk menyesuaikan dan memvisualisasikan model linear, mengananlisis multivariat, dan estimasi kepadatan (density). Beberapa dari fungsi termasuk package glm() untuk menyesuaikan model linear umum, Ida() untuk menganalisis diskriminan linear, kde2d() untuk estimasi kepadatan 2D Kernel, dan stepAIC() untuk seleksi model menggunakan Akaike Information Criterion.
x <- rmvnorm(n = n, mean = mu, sigma = sigma)
d <- data.frame(x)
d## X1 X2
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## 909 0.810393761 5.009494557
## 910 0.459501887 5.460521507
## 911 0.652035985 3.588227818
## 912 5.761517649 4.744090488
## 913 3.945121392 3.967634552
## 914 -1.543166069 -0.881353071
## 915 -2.668842332 -0.203441069
## 916 4.491007495 3.446181852
## 917 0.705621377 2.331727642
## 918 4.310665177 3.846041367
## 919 4.857066660 2.232436201
## 920 3.538809281 2.422431241
## 921 0.998343224 0.569040408
## 922 -0.449655695 1.362602673
## 923 2.164599037 1.008353978
## 924 0.030585539 -0.622116723
## 925 1.234657887 2.218273661
## 926 -1.817220927 1.879695111
## 927 -2.619134992 -2.189272848
## 928 -1.206297173 3.452274159
## 929 2.159383873 2.764792255
## 930 1.031232382 1.282540206
## 931 5.724396631 6.812660391
## 932 -1.626541470 -0.427594446
## 933 1.080920726 3.946389714
## 934 1.686371218 2.718581990
## 935 4.933254482 1.904507895
## 936 0.229482916 0.105368412
## 937 -0.151649735 1.240307762
## 938 -1.329359866 -0.422606392
## 939 1.858513222 0.858375430
## 940 5.576093470 2.290211351
## 941 -0.040137630 -0.275389429
## 942 -2.618707965 0.541293224
## 943 4.498456402 1.399071689
## 944 -0.257927838 -0.100369833
## 945 -1.133927788 3.461035877
## 946 -0.077213401 2.658056333
## 947 0.862429978 1.917443057
## 948 3.737650295 0.866248657
## 949 -1.321416522 0.714429657
## 950 0.293033700 -0.491123981
## 951 1.849931791 2.888068275
## 952 0.359948662 3.192676038
## 953 3.466102661 3.962937235
## 954 0.770163296 0.487263229
## 955 -0.195481461 0.441723598
## 956 1.343873266 0.045078437
## 957 3.713658169 2.541371186
## 958 -1.688053222 2.100863608
## 959 -0.699018041 1.141092408
## 960 3.134976559 3.076749283
## 961 -0.563164173 1.063828274
## 962 3.647676230 2.806313279
## 963 1.979373483 1.814746761
## 964 -0.191328829 1.062900730
## 965 0.691504468 2.232255569
## 966 -0.796480892 2.478193485
## 967 -0.181206250 3.155849760
## 968 3.295755900 2.459883627
## 969 2.571179561 4.014913878
## 970 2.076649378 3.141691672
## 971 4.394712413 4.783027637
## 972 1.148266821 -0.321385176
## 973 4.541154620 4.266722558
## 974 -1.142516433 1.336142290
## 975 0.058409897 0.966677929
## 976 2.856695345 1.389785985
## 977 -0.354709348 0.568430830
## 978 0.975332812 5.421907474
## 979 -0.559898764 -1.373835399
## 980 1.657227350 2.234547880
## 981 0.993608343 1.928292849
## 982 -0.739504863 1.632700670
## 983 0.452094188 1.209386063
## 984 2.575146280 3.165028604
## 985 4.509289099 0.816104443
## 986 4.108578047 4.085592535
## 987 2.968204362 1.515560004
## 988 0.718782643 0.878512393
## 989 0.380063160 3.653205724
## 990 2.209908214 3.216176951
## 991 0.001144748 2.766351068
## 992 -0.153519083 0.168532046
## 993 3.912180515 4.078180540
## 994 1.702980680 2.142721980
## 995 2.206760833 3.961702538
## 996 5.137988495 2.256163366
## 997 2.196996903 5.465947368
## 998 -3.471882095 -0.834231899
## 999 -1.780847675 0.656876922
## 1000 -0.739232156 0.552785941Selanjutnya, menggunakan package ggplot2 untuk memplot data point tersebut
p2 <- ggplot(d, aes(x = X1, y = X2)) +
geom_point(alpha = .5) +
geom_density_2d()
p2
head(x)## [,1] [,2]
## [1,] -1.3922967 0.3812718
## [2,] 2.3032613 5.3532121
## [3,] 0.3243254 1.6474804
## [4,] 4.5941270 0.9380632
## [5,] 3.2442687 2.3019668
## [6,] -1.1183727 -0.7709737mu## [1] 1 2Pertama, menerjemahkan semua data point yang tersimpan sebagai variabel x dengan -mu = [-1, -2], karena pusat distribusi berada di mu = [1,2]. x dikurangi mu menghasilkan y, kolom 1 dikurangi 1 dan kolom 2 dikurangi 2 maka nilainya akan berubah. Serta menyimpan data point terbaru sebagai variabel y.
y <- x - mu
head(y)## [,1] [,2]
## [1,] -2.3922967 -0.6187282
## [2,] 0.3032613 3.3532121
## [3,] -0.6756746 0.6474804
## [4,] 2.5941270 -1.0619368
## [5,] 2.2442687 1.3019668
## [6,] -3.1183727 -2.7709737Tetapi, dikonversi terlebih dahulu ke data.frame agar bisa menampilkan plotnya dan membuat variabel baru yaitu d1.
d1 <- data.frame(y)
d1## X1 X2
## 1 -2.3922966515 -0.618728212
## 2 0.3032613128 3.353212136
## 3 -0.6756746392 0.647480386
## 4 2.5941269922 -1.061936828
## 5 2.2442686531 1.301966752
## 6 -3.1183727049 -2.770973676
## 7 2.2378822955 0.214074108
## 8 0.9202542073 -0.137555309
## 9 -0.6169667041 2.153993314
## 10 -0.7300239811 -1.256668320
## 11 -0.4660789615 -0.110607407
## 12 2.4126825215 0.709948083
## 13 -0.1503354398 0.244549299
## 14 -0.7795004140 1.253586736
## 15 2.5106990079 1.865964879
## 16 0.8732831071 -0.675274492
## 17 -2.1579649770 1.127225298
## 18 -2.3308579598 -1.482746068
## 19 1.3527065911 1.522404354
## 20 -0.5176892962 -2.647181659
## 21 -1.9381066356 1.124397165
## 22 -1.3250965083 -1.603036154
## 23 1.2331033221 1.710099217
## 24 -3.4205178401 -0.263621408
## 25 1.6352939176 0.753433810
## 26 -3.7880181136 -0.684258154
## 27 1.7694754641 1.041372693
## 28 1.5287826813 0.278188567
## 29 1.0096678396 2.861512502
## 30 -0.4145303918 0.196924418
## 31 1.3669067400 1.522129525
## 32 -2.6148728969 0.245450377
## 33 3.7893429670 3.083740963
## 34 -1.7515358030 1.035309493
## 35 -0.1705813759 0.935258116
## 36 1.0665625879 -1.276371982
## 37 1.3130289896 3.233996825
## 38 -2.2515405849 -3.281441750
## 39 0.8706445486 2.319809815
## 40 -1.6782223597 1.336149953
## 41 0.9632078821 2.499603262
## 42 -0.4311262477 1.149350090
## 43 -1.4925918879 -2.301193972
## 44 -0.9083628087 2.708451630
## 45 -0.7384896446 -1.320130752
## 46 1.1051234432 2.449831579
## 47 2.0404697983 1.288865739
## 48 -2.1723509805 -1.626206732
## 49 0.8077668369 0.230857881
## 50 -2.3876994508 -1.190863324
## 51 -0.0005528613 1.984979165
## 52 -0.6751912639 1.936380679
## 53 1.4485882099 1.150705562
## 54 0.1291233547 2.496550811
## 55 1.2532157059 1.646136663
## 56 -0.5390632110 0.150735390
## 57 1.0388076192 1.211131438
## 58 -0.2183209199 1.816921979
## 59 1.7097651678 2.756623009
## 60 -2.0080262220 1.723792490
## 61 0.3120467083 -0.230731680
## 62 1.8162813024 1.800754815
## 63 0.7685521381 2.616657803
## 64 0.5609772284 0.700948958
## 65 2.7167056354 2.333144363
## 66 1.7851952911 0.012769477
## 67 -1.2243050870 -0.178051758
## 68 -1.3258355885 -0.392559620
## 69 2.2032934020 2.423440206
## 70 -0.6798228316 2.259970746
## 71 2.9045493110 2.980497729
## 72 -1.7581362208 -0.805071558
## 73 -0.4049317020 1.063452731
## 74 -1.0575402822 0.234626344
## 75 -0.9040796598 1.560684128
## 76 1.2556525152 1.178235018
## 77 -2.5767920293 0.220276185
## 78 -1.9278270797 -1.678100568
## 79 1.7497916157 1.188940160
## 80 -0.0936855670 2.586496979
## 81 0.2905617233 -0.060777577
## 82 0.6814768679 1.937212237
## 83 -3.1473608726 -1.228411582
## 84 1.5557385781 2.129677088
## 85 3.1231454178 2.835361662
## 86 2.4372564647 1.056248360
## 87 -2.9624395854 -0.567922415
## 88 -3.9473399613 -0.985166423
## 89 2.0662902188 -0.518863661
## 90 -3.6117551964 0.605697601
## 91 -4.4431291309 -0.524485891
## 92 -1.1852131082 -2.136797252
## 93 0.8108235331 -0.164143426
## 94 -0.1948428281 1.535496649
## 95 1.4799506388 2.675271533
## 96 -0.1862281447 -0.086930774
## 97 -2.5025411493 1.168359076
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## 99 -1.7920322251 0.767989685
## 100 -0.1274598048 1.435613156
## 101 4.0441702824 -2.174459971
## 102 -1.9911811960 -2.635927514
## 103 1.2213714674 1.775666283
## 104 -1.8505098549 -3.090241016
## 105 -0.9262680764 -0.046939995
## 106 -1.1656075532 0.866076515
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## 110 -2.3706374961 -0.052588726
## 111 2.6340601526 1.860473031
## 112 0.4979593362 1.420185113
## 113 -1.1334151983 2.993704858
## 114 -4.2867740314 -1.443076588
## 115 0.8064273092 0.117472626
## 116 -3.9687133485 0.915681462
## 117 0.8624176207 1.046270488
## 118 -0.6273444591 1.266697611
## 119 0.0074582947 0.255463493
## 120 0.0228998759 -0.444021447
## 121 4.2244522953 2.541492074
## 122 1.8412546298 1.375734721
## 123 0.5794165392 0.475627872
## 124 -0.2887111267 0.499202690
## 125 -1.7055119200 3.367554119
## 126 0.8682312091 -0.210564378
## 127 0.5965991684 0.949688765
## 128 -5.6026222612 -4.140335046
## 129 -0.2323161386 -0.668143319
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## 143 -1.8709454676 1.605450561
## 144 1.4444415856 3.041089761
## 145 1.3571118259 1.287735600
## 146 -3.2247278440 -1.770204813
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## 151 -2.2636453868 -0.273839661
## 152 1.0884957855 1.734087565
## 153 3.7578159537 2.463488065
## 154 0.4420562902 0.153702638
## 155 1.2798008234 2.161055863
## 156 -2.9733201536 0.986367504
## 157 -1.0578693918 -0.209042547
## 158 -2.5425781262 -0.551958606
## 159 1.0200999775 3.273589743
## 160 -2.9074816821 -0.750467214
## 161 1.8380991968 2.741568892
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## 165 -1.4870811335 1.052206466
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## 167 0.6229344264 -1.034431081
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## 169 -1.8767129892 4.122255469
## 170 -3.2742759067 -1.089085894
## 171 -0.6442511379 0.305002998
## 172 0.7493180000 -1.290271140
## 173 -0.9951048785 0.037005288
## 174 -0.2040957514 1.059359411
## 175 2.3397345186 3.283315982
## 176 -1.8316759067 -1.578357628
## 177 -0.3188252882 -2.404927896
## 178 -0.8996610826 0.194564942
## 179 0.6124753708 3.582170830
## 180 -2.2537856670 -1.379797634
## 181 -0.9987055848 0.206557536
## 182 0.4898842689 2.389645851
## 183 -0.0777082370 1.645239889
## 184 1.5671956819 -0.532186465
## 185 -0.0852975452 1.642602984
## 186 1.7651237962 -0.971003996
## 187 4.4058429337 1.801158277
## 188 1.1540846601 3.395578716
## 189 -5.0047874560 -3.639636309
## 190 -2.7237666875 -2.133747634
## 191 0.5743619819 0.412618903
## 192 0.7597713820 -0.854401275
## 193 0.0067048857 1.235980003
## 194 0.5449905861 -0.119026273
## 195 3.2448588068 0.841899824
## 196 -0.5254201159 -1.031643617
## 197 0.5430303784 -0.326064143
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## 199 2.8254902504 0.660510687
## 200 1.4212953704 4.443834487
## 201 1.4487073096 -1.451399622
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## 203 -2.7349662234 -1.900238150
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## 206 -2.0859581650 -1.755311786
## 207 3.7418815583 2.435559141
## 208 1.2456765063 -2.029597208
## 209 -0.3280492363 0.870981295
## 210 -1.9899401514 -1.534438077
## 211 -0.6648880775 -2.722531394
## 212 1.2022995429 0.499952337
## 213 -2.5554716309 -1.496083405
## 214 0.1205012612 1.016499143
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## 216 -3.5207277611 1.688422611
## 217 -2.8812008719 -0.455220026
## 218 -2.6716227815 -0.686455063
## 219 -1.2967470182 -0.015669997
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## 222 2.2137423865 1.311125897
## 223 3.2688716610 2.139720923
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## 232 1.5448119430 -0.027616152
## 233 -0.1638882691 0.669249954
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## 235 5.2797090588 6.201427392
## 236 2.4247247689 3.300579195
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## 238 -1.7167121631 -0.896814340
## 239 0.2181879068 1.541399913
## 240 -0.6924359254 1.218419809
## 241 3.4143425224 0.384717038
## 242 0.1867517357 1.185898943
## 243 -2.3392214033 -1.140553067
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## 245 1.3848725482 1.129516885
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## 252 -3.2369700670 -4.070597809
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## 816 0.4860263627 0.319271555
## 817 1.1305416985 0.881492480
## 818 0.5257162783 1.875972971
## 819 -0.2924182804 -0.011557246
## 820 1.9305118741 2.923768037
## 821 -4.0771429741 -2.887475856
## 822 0.9775147505 0.854590179
## 823 1.6526988184 0.416017050
## 824 2.6046243450 -0.780288502
## 825 0.2487088484 2.166527059
## 826 -2.1651605123 -1.686457548
## 827 -1.3412920360 -1.837404387
## 828 1.6119397747 0.250813485
## 829 -4.1355728658 -0.568301578
## 830 0.9408086958 0.662111373
## 831 -0.8214720260 0.095686817
## 832 -1.9602918593 1.218635600
## 833 0.1424773335 1.578111824
## 834 2.7700376152 1.909349492
## 835 0.7014235058 1.075835514
## 836 2.0689851904 -1.053252540
## 837 -0.6182217379 0.584101176
## 838 -3.5636021957 -0.663689851
## 839 -1.9267311388 1.048202058
## 840 -0.8974181170 0.149922535
## 841 0.2786778631 2.518455503
## 842 -3.0409889430 -2.627431279
## 843 0.7721371981 0.209516131
## 844 -6.4098257590 -3.419488261
## 845 3.1250443788 4.890456251
## 846 2.7059167374 2.070501748
## 847 -2.4450921211 -0.235372633
## 848 0.8395636349 0.257659083
## 849 -3.4225910018 0.181368364
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## 852 -2.5518299926 -0.868787678
## 853 -1.3176016135 -1.483906473
## 854 0.7820777242 -0.964608842
## 855 1.3300991137 2.704072591
## 856 -3.1471462247 -2.618971852
## 857 -0.3862775679 1.295048067
## 858 -0.0199853064 -1.647699983
## 859 -2.0179692489 -0.431176921
## 860 -3.1512465364 -1.692437382
## 861 4.1543730187 3.946379116
## 862 1.5662128914 2.701807503
## 863 0.0181313833 0.752697867
## 864 -1.2977113847 1.158010333
## 865 0.6813740115 -0.162727307
## 866 1.7357996394 2.551977027
## 867 5.6468981587 5.568760211
## 868 -5.1449108280 -2.754172380
## 869 0.0385152718 2.170814053
## 870 -0.1281875158 0.960181414
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## 874 -0.3565172569 0.604030287
## 875 -1.2517465380 1.837815156
## 876 1.7796310260 3.211541748
## 877 0.2154597549 2.924356248
## 878 -0.2830514820 -2.020405710
## 879 1.6257105954 1.638640967
## 880 -1.5928885098 -1.346931603
## 881 -1.7598152243 0.584629945
## 882 -3.7784752431 -0.362258366
## 883 0.3008964646 0.999123671
## 884 -0.7737728179 -1.547796291
## 885 -0.8021754823 -0.794976616
## 886 -2.6221312118 0.462270996
## 887 -1.3435402151 -1.548146539
## 888 -0.8938868446 2.138444653
## 889 2.3568705007 2.839018491
## 890 -0.4966792145 -0.777806635
## 891 3.1696479749 4.580021368
## 892 -2.5956535295 0.732759750
## 893 -0.3821589057 1.158573747
## 894 1.2683208036 3.705464559
## 895 1.7857259831 0.481770624
## 896 -2.3320558724 -2.940104680
## 897 -1.6125254172 -0.930722041
## 898 -2.1060333805 -0.852924629
## 899 1.7670388817 3.145287327
## 900 1.7186154721 1.365577830
## 901 -0.5331085146 1.581924812
## 902 -1.7120684565 0.381117802
## 903 0.7410862989 1.922208145
## 904 -4.5686875096 -0.328693931
## 905 2.1236030359 1.703483624
## 906 -3.1590411254 0.456645463
## 907 4.6695051359 2.968416050
## 908 3.6719178991 1.140666400
## 909 -0.1896062388 4.009494557
## 910 -1.5404981128 3.460521507
## 911 -0.3479640152 2.588227818
## 912 3.7615176486 2.744090488
## 913 2.9451213921 2.967634552
## 914 -3.5431660693 -2.881353071
## 915 -3.6688423321 -1.203441069
## 916 2.4910074946 1.446181852
## 917 -0.2943786233 1.331727642
## 918 2.3106651774 1.846041367
## 919 3.8570666605 1.232436201
## 920 1.5388092811 0.422431241
## 921 -0.0016567758 -0.430959592
## 922 -2.4496556946 -0.637397327
## 923 1.1645990372 0.008353978
## 924 -1.9694144609 -2.622116723
## 925 0.2346578871 1.218273661
## 926 -3.8172209268 -0.120304889
## 927 -3.6191349919 -3.189272848
## 928 -3.2062971734 1.452274159
## 929 1.1593838729 1.764792255
## 930 -0.9687676178 -0.717459794
## 931 4.7243966312 5.812660391
## 932 -3.6265414700 -2.427594446
## 933 0.0809207255 2.946389714
## 934 -0.3136287820 0.718581990
## 935 3.9332544817 0.904507895
## 936 -1.7705170844 -1.894631588
## 937 -1.1516497349 0.240307762
## 938 -3.3293598656 -2.422606392
## 939 0.8585132219 -0.141624570
## 940 3.5760934704 0.290211351
## 941 -1.0401376302 -1.275389429
## 942 -4.6187079655 -1.458706776
## 943 3.4984564016 0.399071689
## 944 -2.2579278376 -2.100369833
## 945 -2.1339277881 2.461035877
## 946 -2.0772134007 0.658056333
## 947 -0.1375700224 0.917443057
## 948 1.7376502949 -1.133751343
## 949 -2.3214165222 -0.285570343
## 950 -1.7069662997 -2.491123981
## 951 0.8499317911 1.888068275
## 952 -1.6400513375 1.192676038
## 953 2.4661026612 2.962937235
## 954 -1.2298367040 -1.512736771
## 955 -1.1954814605 -0.558276402
## 956 -0.6561267339 -1.954921563
## 957 2.7136581689 1.541371186
## 958 -3.6880532225 0.100863608
## 959 -1.6990180410 0.141092408
## 960 1.1349765591 1.076749283
## 961 -1.5631641727 0.063828274
## 962 1.6476762300 0.806313279
## 963 0.9793734826 0.814746761
## 964 -2.1913288294 -0.937099270
## 965 -0.3084955324 1.232255569
## 966 -2.7964808920 0.478193485
## 967 -1.1812062505 2.155849760
## 968 1.2957558998 0.459883627
## 969 1.5711795605 3.014913878
## 970 0.0766493779 1.141691672
## 971 3.3947124129 3.783027637
## 972 -0.8517331793 -2.321385176
## 973 3.5411546202 3.266722558
## 974 -3.1425164327 -0.663857710
## 975 -0.9415901029 -0.033322071
## 976 0.8566953455 -0.610214015
## 977 -1.3547093481 -0.431569170
## 978 -1.0246671880 3.421907474
## 979 -1.5598987637 -2.373835399
## 980 -0.3427726500 0.234547880
## 981 -0.0063916571 0.928292849
## 982 -2.7395048627 -0.367299330
## 983 -0.5479058118 0.209386063
## 984 0.5751462796 1.165028604
## 985 3.5092890986 -0.183895557
## 986 2.1085780466 2.085592535
## 987 1.9682043616 0.515560004
## 988 -1.2812173565 -1.121487607
## 989 -0.6199368397 2.653205724
## 990 0.2099082142 1.216176951
## 991 -0.9988552517 1.766351068
## 992 -2.1535190831 -1.831467954
## 993 2.9121805147 3.078180540
## 994 -0.2970193197 0.142721980
## 995 1.2067608330 2.961702538
## 996 3.1379884947 0.256163366
## 997 1.1969969032 4.465947368
## 998 -5.4718820950 -2.834231899
## 999 -2.7808476747 -0.343123078
## 1000 -2.7392321559 -1.447214059Untuk melihat semua jumlah data yaitu dengan dim dan hasilnya ada 1000 data
dim(d1)## [1] 1000 2Sehingga grafiknya dibawah berbeda.
p2 <- ggplot(d1, aes(x = X1, y = X2)) +
geom_point(alpha = .5) +
geom_density_2d()
p2
Pada kode di bawah ini, ggplot() menginisialisasi objek plot, dan aes() mengspesifikasikan pemetaan antara variabel x dan y dan sifat estetika pada plot. geom_point() menambahkan layer scatter plot ke objek plot.
Apabila ingin membuat data yang bermanfaat atau yang memiliki nilai maka beri bentuk dalam data.frame, karena jika matriks saja itu masih data mentah, belum memiliki nilai.
x <- 1:1000
ggplot(d1, aes(x = x, y = d$X1)) +
geom_point()
Berikutnya, menghitung eigen vectors dari sigma.
E <- eigen(sigma)
E$vectors## [,1] [,2]
## [1,] -0.7882054 0.6154122
## [2,] -0.6154122 -0.7882054Misalnya, matriks dimensinya berbeda-beda tetapi isinya sama. Untuk mengidentifikasi suatu matriks maka menggunakan E$values karena mewakili matriks tersebut. Matriks memiliki identitas dan bisa dibanding-bandingkan.
Vektor eigen dari matriks sigma disimpan dalam E$vectors dalam bentuk sebuah matriks. Setiap vektor kolom adalah sebuah vektor eigen. Dan mengambil perkalian matriks, kumpulan data disimpan menjadi matriks perkalian 10000x2 dengan matriks 2x2, transpose invers dari matriks E.
y dikali dengan identitas (transpose) akan kembali ke y
y <- y %*% t(inv(E$vectors))
head(y)## [,1] [,2]
## [1,] 2.2663941 -0.9845636
## [2,] -2.3026399 -2.4563893
## [3,] 0.1341031 -0.9261660
## [4,] -1.3911761 2.4334818
## [5,] -2.5701910 0.3549331
## [6,] 4.1632094 0.2650119Seperti dengan kumpulan data x sebelumnya diberi bentuk visual dengan “data frame” dengan fungsi data.frame() agar mudah dipahami dan lebih bernilai.
dd <- data.frame(y)
dd## X1 X2
## 1 2.266394131 -0.9845636263
## 2 -2.302639908 -2.4563893326
## 3 0.134103091 -0.9261659851
## 4 -1.391176117 2.4334818101
## 5 -2.570190998 0.3549330547
## 6 4.163209364 0.2650118879
## 7 -1.895654819 1.2084857126
## 8 -0.640696156 0.6747575181
## 9 -0.839297273 -2.0774780911
## 10 1.348777901 0.5412471348
## 11 0.435435121 -0.1996493242
## 12 -2.338600207 0.9252093416
## 13 -0.032003413 -0.2852733533
## 14 -0.157066117 -1.4677979571
## 15 -3.127284186 0.0743511565
## 16 -0.272754328 1.0696841150
## 17 1.007211523 -2.2165231073
## 18 2.749694958 -0.2657299313
## 19 -2.003116922 -0.3674952413
## 20 2.037153435 1.7679306706
## 21 0.835658450 -2.0789904496
## 22 2.030976298 0.4480412462
## 23 -2.024354685 -0.5890426652
## 24 2.858306602 -1.8972406156
## 25 -1.752619928 0.4125192165
## 26 3.406837307 -1.7918565997
## 27 -2.035583657 0.2681411842
## 28 -1.376195467 0.7215617867
## 29 -2.556835417 -1.6340978043
## 30 0.205545418 -0.4103239624
## 31 -2.014140423 -0.3585396742
## 32 1.910003883 -1.8026900309
## 33 -4.884552582 -0.0986134727
## 34 0.743427945 -1.8939530940
## 35 -0.441116096 -0.8421533964
## 36 -0.055175531 1.6624189794
## 37 -3.025177725 -1.7409998178
## 38 3.794115857 1.2008246714
## 39 -2.113886054 -1.2926814298
## 40 0.500500998 -2.0859591931
## 41 -2.297492060 -1.3774309975
## 42 -0.367508025 -1.1712443505
## 43 2.592651914 0.8952543347
## 44 -0.950837696 -2.6938338722
## 45 1.394506139 0.5860586959
## 46 -2.378720576 -1.2508641174
## 47 -2.401493108 0.2398390413
## 48 2.713046339 -0.0551063252
## 49 -0.778758974 0.3151461365
## 50 2.614869526 -0.5307744459
## 51 -1.221144646 -1.5649116137
## 52 -0.659482886 -1.9417867327
## 53 -1.849943361 -0.0155135125
## 54 -1.638183582 -1.8883308413
## 55 -2.000844038 -0.5262496252
## 56 0.332128156 -0.4505565368
## 57 -1.564138892 -0.3153254955
## 58 -0.946074234 -1.5664651474
## 59 -3.044105665 -1.1205748913
## 60 0.521894246 -2.5944664732
## 61 -0.103961820 0.3739013199
## 62 -2.539809304 -0.3016030513
## 63 -2.216100138 -1.5894875452
## 64 -0.873537850 -0.2072595459
## 65 -3.577167689 -0.1671032602
## 66 -1.414959132 1.0885660081
## 67 1.074579156 -0.6131109348
## 68 1.286616806 -0.5065177819
## 69 -3.228062538 -0.5542350921
## 70 -0.854973537 -2.1996925077
## 71 -4.123616262 -0.5617494134
## 72 1.881223400 -0.4474167155
## 73 -0.335292425 -1.0874191417
## 74 0.689167087 -0.8357569630
## 75 -0.247863563 -1.7865213815
## 76 -1.714812359 -0.1559473613
## 77 1.895480841 -1.7594121649
## 78 2.552247371 0.1362796733
## 79 -2.110883562 0.1397140229
## 80 -1.517918348 -2.0963462311
## 81 -0.191619068 0.2267204492
## 82 -1.729327838 -1.1075320380
## 83 3.236746448 -0.9686836184
## 84 -2.536870894 -0.7212025497
## 85 -4.206596395 -0.3128256624
## 86 -2.571086941 0.6673766843
## 87 2.684517385 -1.3754819552
## 88 3.717598277 -1.6527276751
## 89 -1.309346159 1.6805913899
## 90 2.474051394 -2.7001323917
## 91 3.824873573 -2.3209532856
## 92 2.249202539 0.9548405999
## 93 -0.538079651 0.6283694435
## 94 -0.791387209 -1.3301954673
## 95 -2.812899911 -1.1978838827
## 96 0.200284296 -0.0460877652
## 97 1.253494107 -2.4610013590
## 98 1.074689063 1.2772790584
## 99 0.939859319 -1.7081721599
## 100 -0.783029354 -1.2099984195
## 101 -1.849447801 4.2027529490
## 102 3.191641828 0.8522551857
## 103 -2.055458346 -0.6479429096
## 104 3.360353987 1.2969184202
## 105 0.758976983 -0.5330383244
## 106 0.385744152 -1.3999753413
## 107 0.303432762 -0.2016559156
## 108 1.414671342 0.0325158518
## 109 2.410588840 -0.8792228582
## 110 1.900913115 -1.4174685406
## 111 -3.221138361 0.1545978157
## 112 -1.266493517 -0.8129473763
## 113 -0.948998498 -3.0571720072
## 114 4.266945564 -1.5006922632
## 115 -0.707924481 0.4036926495
## 116 2.564639899 -3.1641397620
## 117 -1.323649893 -0.2939337561
## 118 -0.285064861 -1.3844933881
## 119 -0.163094021 -0.1967677891
## 120 0.255206413 0.3640729834
## 121 -4.893801534 0.5965616446
## 122 -2.297930861 0.0487689899
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## 124 -0.079651750 -0.5711506282
## 125 -0.728140149 -3.7039173360
## 126 -0.554760673 0.7002880754
## 127 -1.054692772 -0.3813954386
## 128 6.964030085 -0.1844875411
## 129 0.594296401 0.3836640101
## 130 0.318969849 -0.9674313512
## 131 -1.558977239 -1.6860197737
## 132 0.622319125 -0.5029550731
## 133 1.905178734 -2.4246555107
## 134 -2.410975842 -1.7159220754
## 135 -0.926617343 -2.1836745274
## 136 1.460457045 -1.3238981002
## 137 -1.269179056 -1.7873718247
## 138 3.380223546 -0.6297960082
## 139 -3.482351872 -1.1199406727
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## 141 -0.907763052 -1.2396504225
## 142 4.340886214 -2.1636927059
## 143 0.486675518 -2.4168275506
## 144 -3.010040486 -1.5080765050
## 145 -1.862171135 -0.1798170172
## 146 3.631153685 -0.5892518256
## 147 3.455410382 -1.2900652946
## 148 2.689731465 -0.8105412191
## 149 0.532955871 -0.6971120323
## 150 3.551124508 -1.0550233962
## 151 1.952741879 -1.1772330995
## 152 -1.925136960 -0.6969436552
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## 154 -0.443021653 0.1508975834
## 155 -2.338686136 -0.9157509341
## 156 1.736564514 -2.6072777594
## 157 0.962465745 -0.4862572677
## 158 2.343755976 -1.1296768482
## 159 -2.818655450 -1.9524792622
## 160 2.753539565 -1.1977773870
## 161 -3.135994757 -1.0297308260
## 162 2.309513533 -0.8369493197
## 163 0.786777139 -0.6842478195
## 164 1.613005550 0.8254629975
## 165 0.524584732 -1.7445227475
## 166 2.794467120 -0.0954449213
## 167 0.145601214 1.1987056573
## 168 5.199794343 0.8298168012
## 169 -1.057650961 -4.4041362738
## 170 3.251038838 -1.1566059460
## 171 0.320099682 -0.6368850390
## 172 0.203432090 1.4781381780
## 173 0.761573573 -0.6415674616
## 174 -0.491073335 -0.9605958682
## 175 -3.864784221 -1.1480263277
## 176 2.415077470 0.1168343505
## 177 1.731321818 1.6993682751
## 178 0.589380118 -0.7070195609
## 179 -2.687268086 -2.4465617139
## 180 2.625590435 -0.2994432171
## 181 0.660067145 -0.7774253850
## 182 -1.856746680 -1.5820510991
## 183 -0.951250661 -1.3446096284
## 184 -0.907758113 1.3839436252
## 185 -0.943645943 -1.3472017586
## 186 -0.793712464 1.8516293679
## 187 -4.581164164 1.2917267846
## 188 -2.999336409 -1.9661758246
## 189 6.184677323 -0.2112261707
## 190 3.460022067 0.0055922157
## 191 -0.706645950 0.0282409125
## 192 -0.073046960 1.1410163181
## 193 -0.765922013 -0.9700798936
## 194 -0.356314323 0.4292110165
## 195 -3.075730795 1.3333357084
## 196 1.049025072 0.4897971566
## 197 -0.227355644 0.5911930564
## 198 6.780967338 -1.4619446160
## 199 -2.633553128 1.2182230826
## 200 -3.855062745 -2.6279719923
## 201 -0.248669933 2.0355532451
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## 203 3.325145015 -0.1853535610
## 204 0.905164629 -1.1656969108
## 205 3.135762710 -0.7665537380
## 206 2.724403879 0.0998221740
## 207 -4.448244234 0.3830786349
## 208 0.267189904 2.3663440919
## 209 -0.277442331 -0.8883977000
## 210 2.512793580 -0.0151810271
## 211 2.199547462 1.7367338143
## 212 -1.255335813 0.3458446667
## 213 2.934944636 -0.3934473657
## 214 -0.720545734 -0.7270522067
## 215 1.139861251 -1.1580516535
## 216 1.735980884 -3.4975227393
## 217 2.551126163 -1.4143192954
## 218 2.528240438 -1.1030816650
## 219 1.031746561 -0.7856827718
## 220 4.356608354 -1.6170866778
## 221 -2.695126421 0.1187324704
## 222 -2.551766677 0.3289275301
## 223 -3.893352808 0.3251638617
## 224 0.774079021 -1.2298214010
## 225 2.304443123 -3.3744277738
## 226 1.676063467 -1.0345364192
## 227 1.980433907 -0.3817234629
## 228 3.256163060 -0.9615933680
## 229 0.610986879 -1.5638706061
## 230 3.423648710 1.8263580331
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## 976 -0.299718776 1.0081947823
## 977 1.333382214 -0.4935395065
## 978 -1.298235389 -3.3277587845
## 979 2.690407980 0.9110892294
## 980 0.125831638 -0.3958183888
## 981 -0.566244815 -0.7356189776
## 982 2.385333128 -1.3964174116
## 983 0.303003602 -0.5022271606
## 984 -1.170306254 -0.5643298401
## 985 -2.652869187 2.3046068384
## 986 -2.945491798 -0.3462307060
## 987 -1.868631306 0.8048897963
## 988 1.700039657 0.0954858276
## 989 -1.144177609 -2.4727878857
## 990 -0.913900942 -0.8294172109
## 991 -0.299730871 -2.0069552387
## 992 2.824523197 0.1182710662
## 993 -4.189746413 -0.6340472004
## 994 0.146279394 -0.2952835574
## 995 -2.773843357 -1.5917747005
## 996 -2.631025665 1.7292470757
## 997 -3.691878010 -2.7834375003
## 998 6.057188151 -1.1335060516
## 999 2.403041397 -1.4409161363
## 1000 3.049710889 -0.5450549206Setelah itu, maka akan di plot dengan menggunakan kode di bawah ini:
p3 <- ggplot(dd, aes(x = X1, y = X2)) +
geom_point(alpha = .5) +
geom_density_2d()
p3
library(ggplot2)
# membuat sampel data
x <- c(10, 20, 30, 40, 50)
y <- c(20, 30, 40, 50, 60)
# membuat objek ggplot dan menambahkan layer scatter plot
ggplot(data = data.frame(x, y), aes(x = x, y = y)) +
geom_point()
Daftar PustakaYoshida, Ruriko. (2021). Linear Algebra and Its Applications with R. CRC Press.