Vector Spaces and Introductory Example from Data Science
Wafiy Anwarul Hikam
2023-05-30
Oke, sekarang kita akan membahas tentang Ruang Vektor (Vector Spaces)
Sebuah ruang vektor adalah dasar dalam banyak bidang. Sebagai contoh, dalam statistik, kita mengasumsikan properti dari ruang vektor agar dapat melakukan analisis statistik pada kumpulan data. Pada bab ini, kita fokus pada ruang vektor dan properti dasarnya. Pertama, kita mendefinisikan ruang vektor dan kemudian untuk aplikasinya, kita mengasumsikan “ruang Euklides” sebagai ruang vektor.
Contoh Pengantar Data Science
Dalam ilmu data, terdapat tiga sub-area dasar dari model statistik: pembelajaran terawasi (supervised learning), pembelajaran tak terawasi (unsupervised learning), dan pembelajaran penguatan (reinforcement learning). Pada bab ini, kita fokus pada pembelajaran tak terawasi, terutama klasterisasi. Model pembelajaran tak terawasi adalah metode statistik deskriptif, berbeda dengan pembelajaran terawasi yang merupakan metode statistik inferensial untuk memprediksi hasil atau parameter berdasarkan dataset. Salah satu area utama dalam pembelajaran tak terawasi adalah klasterisasi. Metode klasterisasi digunakan untuk mengelompokkan observasi dalam dataset berdasarkan “kemiripan” mereka. Pengukuran “kemiripan” ini didefinisikan sebagai “produk dot” vektor dalam ruang vektor.
Dalam bagian ini, kita akan mempertimbangkan dataset kejahatan dari Amerika Serikat, yaitu dataset “USArrests” dari R. Data ini berisi informasi tentang penangkapan yang dilakukan di Amerika Serikat, yaitu jumlah penangkapan rata-rata per 100.000 penduduk untuk kasus serangan, pembunuhan, pemerkosaan, dan persentase populasi yang tinggal di daerah perkotaan di masing-masing dari 50 negara bagian pada tahun 1973.
Di sini, kita akan menggunakan fungsi
hclust() dari R untuk melihat
pengelompokan (klasterisasi), yaitu mengelompokkan berdasarkan kemiripan
antara setiap observasi. Pertama, kita akan mengunggah data “USArrests”
dari R.
df <-USArrestsKita dapat melihat 6 observasi pertama dengan menggunakan fungsi head() seperti berikut:
head(df)## Murder Assault UrbanPop Rape
## Alabama 13.2 236 58 21.2
## Alaska 10.0 263 48 44.5
## Arizona 8.1 294 80 31.0
## Arkansas 8.8 190 50 19.5
## California 9.0 276 91 40.6
## Colorado 7.9 204 78 38.7Kemudian kita membersihkan data dengan menghapus nilai-nilai yang hilang menggunakan fungsi na.omit().
df <-na.omit(df)
df## Murder Assault UrbanPop Rape
## Alabama 13.2 236 58 21.2
## Alaska 10.0 263 48 44.5
## Arizona 8.1 294 80 31.0
## Arkansas 8.8 190 50 19.5
## California 9.0 276 91 40.6
## Colorado 7.9 204 78 38.7
## Connecticut 3.3 110 77 11.1
## Delaware 5.9 238 72 15.8
## Florida 15.4 335 80 31.9
## Georgia 17.4 211 60 25.8
## Hawaii 5.3 46 83 20.2
## Idaho 2.6 120 54 14.2
## Illinois 10.4 249 83 24.0
## Indiana 7.2 113 65 21.0
## Iowa 2.2 56 57 11.3
## Kansas 6.0 115 66 18.0
## Kentucky 9.7 109 52 16.3
## Louisiana 15.4 249 66 22.2
## Maine 2.1 83 51 7.8
## Maryland 11.3 300 67 27.8
## Massachusetts 4.4 149 85 16.3
## Michigan 12.1 255 74 35.1
## Minnesota 2.7 72 66 14.9
## Mississippi 16.1 259 44 17.1
## Missouri 9.0 178 70 28.2
## Montana 6.0 109 53 16.4
## Nebraska 4.3 102 62 16.5
## Nevada 12.2 252 81 46.0
## New Hampshire 2.1 57 56 9.5
## New Jersey 7.4 159 89 18.8
## New Mexico 11.4 285 70 32.1
## New York 11.1 254 86 26.1
## North Carolina 13.0 337 45 16.1
## North Dakota 0.8 45 44 7.3
## Ohio 7.3 120 75 21.4
## Oklahoma 6.6 151 68 20.0
## Oregon 4.9 159 67 29.3
## Pennsylvania 6.3 106 72 14.9
## Rhode Island 3.4 174 87 8.3
## South Carolina 14.4 279 48 22.5
## South Dakota 3.8 86 45 12.8
## Tennessee 13.2 188 59 26.9
## Texas 12.7 201 80 25.5
## Utah 3.2 120 80 22.9
## Vermont 2.2 48 32 11.2
## Virginia 8.5 156 63 20.7
## Washington 4.0 145 73 26.2
## West Virginia 5.7 81 39 9.3
## Wisconsin 2.6 53 66 10.8
## Wyoming 6.8 161 60 15.6Kemudian kita menggunakan fungsi scale() untuk “standarisasi” setiap pengukuran. Alasan mengapa kita menerapkan fungsi ini adalah karena beberapa pengukuran memiliki nilai yang jauh lebih besar daripada yang lain. Misalnya, seorang siswa mengambil tes ACT dan SAT. Skor SAT memiliki skor maksimum 1600 sedangkan ACT memiliki skor maksimum 36. Jika siswa ini ingin tahu tes mana di mana dia atau dia lebih baik, kita tidak bisa hanya membandingkan skor-skornya. Jika kita menghitung seberapa jauh skor siswa untuk SAT atau ACT dari rerata, maka kita dapat membandingkan skor siswa pada ACT dan SAT. Fungsi scale() menghitung seberapa jauh setiap observasi dari rerata.
df <-scale(df)
df## Murder Assault UrbanPop Rape
## Alabama 1.24256408 0.78283935 -0.52090661 -0.003416473
## Alaska 0.50786248 1.10682252 -1.21176419 2.484202941
## Arizona 0.07163341 1.47880321 0.99898006 1.042878388
## Arkansas 0.23234938 0.23086801 -1.07359268 -0.184916602
## California 0.27826823 1.26281442 1.75892340 2.067820292
## Colorado 0.02571456 0.39885929 0.86080854 1.864967207
## Connecticut -1.03041900 -0.72908214 0.79172279 -1.081740768
## Delaware -0.43347395 0.80683810 0.44629400 -0.579946294
## Florida 1.74767144 1.97077766 0.99898006 1.138966691
## Georgia 2.20685994 0.48285493 -0.38273510 0.487701523
## Hawaii -0.57123050 -1.49704226 1.20623733 -0.110181255
## Idaho -1.19113497 -0.60908837 -0.79724965 -0.750769945
## Illinois 0.59970018 0.93883125 1.20623733 0.295524916
## Indiana -0.13500142 -0.69308401 -0.03730631 -0.024769429
## Iowa -1.28297267 -1.37704849 -0.58999237 -1.060387812
## Kansas -0.41051452 -0.66908525 0.03177945 -0.345063775
## Kentucky 0.43898421 -0.74108152 -0.93542116 -0.526563903
## Louisiana 1.74767144 0.93883125 0.03177945 0.103348309
## Maine -1.30593210 -1.05306531 -1.00450692 -1.434064548
## Maryland 0.80633501 1.55079947 0.10086521 0.701231086
## Massachusetts -0.77786532 -0.26110644 1.34440885 -0.526563903
## Michigan 0.99001041 1.01082751 0.58446551 1.480613993
## Minnesota -1.16817555 -1.18505846 0.03177945 -0.676034598
## Mississippi 1.90838741 1.05882502 -1.48810723 -0.441152078
## Missouri 0.27826823 0.08687549 0.30812248 0.743936999
## Montana -0.41051452 -0.74108152 -0.86633540 -0.515887425
## Nebraska -0.80082475 -0.82507715 -0.24456358 -0.505210947
## Nevada 1.01296983 0.97482938 1.06806582 2.644350114
## New Hampshire -1.30593210 -1.36504911 -0.65907813 -1.252564419
## New Jersey -0.08908257 -0.14111267 1.62075188 -0.259651949
## New Mexico 0.82929443 1.37080881 0.30812248 1.160319648
## New York 0.76041616 0.99882813 1.41349461 0.519730957
## North Carolina 1.19664523 1.99477641 -1.41902147 -0.547916860
## North Dakota -1.60440462 -1.50904164 -1.48810723 -1.487446939
## Ohio -0.11204199 -0.60908837 0.65355127 0.017936483
## Oklahoma -0.27275797 -0.23710769 0.16995096 -0.131534211
## Oregon -0.66306820 -0.14111267 0.10086521 0.861378259
## Pennsylvania -0.34163624 -0.77707965 0.44629400 -0.676034598
## Rhode Island -1.00745957 0.03887798 1.48258036 -1.380682157
## South Carolina 1.51807718 1.29881255 -1.21176419 0.135377743
## South Dakota -0.91562187 -1.01706718 -1.41902147 -0.900240639
## Tennessee 1.24256408 0.20686926 -0.45182086 0.605142783
## Texas 1.12776696 0.36286116 0.99898006 0.455672088
## Utah -1.05337842 -0.60908837 0.99898006 0.178083656
## Vermont -1.28297267 -1.47304350 -2.31713632 -1.071064290
## Virginia 0.16347111 -0.17711080 -0.17547783 -0.056798864
## Washington -0.86970302 -0.30910395 0.51537975 0.530407436
## West Virginia -0.47939280 -1.07706407 -1.83353601 -1.273917376
## Wisconsin -1.19113497 -1.41304662 0.03177945 -1.113770203
## Wyoming -0.22683912 -0.11711392 -0.38273510 -0.601299251
## attr(,"scaled:center")
## Murder Assault UrbanPop Rape
## 7.788 170.760 65.540 21.232
## attr(,"scaled:scale")
## Murder Assault UrbanPop Rape
## 4.355510 83.337661 14.474763 9.366385Kemudian kita menggunakan “jarak Euclidean” sebagai pengukuran seberapa jauh dua observasi berbeda satu sama lain. Di sini, “jarak Euclidean” adalah akar kuadrat dari produk dalam (inner product) dari dua vektor dalam R4 untuk data “USArrests” (kita akan membahas produk dalam dalam Bab 5). Kita menyimpan jarak antara setiap pasangan observasi dalam data “USArrests” dalam bentuk matriks 50x50 menggunakan fungsi dist().
d <-dist(df,method="euclidean")
d## Alabama Alaska Arizona Arkansas California Colorado
## Alaska 2.7037541
## Arizona 2.2935197 2.7006429
## Arkansas 1.2898102 2.8260386 2.7177583
## California 3.2631104 3.0125415 1.3104842 3.7636409
## Colorado 2.6510673 2.3265187 1.3650307 2.8310512 1.2876185
## Connecticut 3.2152975 4.7399125 3.2628575 2.6076395 4.0663898 3.3279920
## Delaware 2.0192927 3.6213633 1.9093696 1.8003239 3.0737852 2.5547456
## Florida 2.2981353 2.9967642 1.7493928 3.3721968 2.0250039 2.4458600
## Georgia 1.1314351 2.8194388 2.7871963 2.2117614 3.3780585 2.8649105
## Hawaii 3.3885300 4.5301340 3.2621208 2.9723097 3.6589083 2.8233524
## Idaho 2.9146623 4.0580555 3.5210071 1.7687255 4.4879436 3.4767685
## Illinois 1.8734993 3.2670626 1.0825512 2.4626424 1.9117469 1.7898322
## Indiana 2.0761411 3.3655952 2.6407486 1.4450503 3.4061273 2.3655622
## Iowa 3.4878952 4.7251910 4.1157513 2.4252661 4.9708591 3.9406898
## Kansas 2.2941096 3.6808173 2.7762838 1.5718411 3.6071725 2.6272281
## Kentucky 1.8475879 3.5440903 3.3567681 1.0598104 4.2463809 3.2274013
## Louisiana 0.7722224 2.9631431 2.2178519 2.0254276 3.0176625 2.6546743
## Maine 3.4851115 4.8322605 4.2961903 2.3621893 5.2699843 4.2713441
## Maryland 1.2896460 2.2777590 1.2117356 2.0582244 2.2312581 1.9667562
## Massachusetts 2.9874810 4.3729925 2.5162281 2.6881270 3.2156499 2.6522793
## Michigan 1.8814771 2.1154937 1.1940906 2.5895050 1.5146739 1.2363108
## Minnesota 3.2314338 4.4266606 3.5388450 2.3300992 4.3123134 3.3283853
## Mississippi 1.2831907 3.2554326 3.4551406 1.9318631 4.4200736 3.8491042
## Missouri 1.6309686 2.5360573 1.5958731 1.6717500 2.2891751 1.3127406
## Montana 2.3317271 3.6575988 3.3270869 1.2290066 4.2494176 3.1845338
## Nebraska 2.6625170 3.9136902 3.1641791 1.7240495 4.0197242 3.0034613
## Nevada 3.1024305 2.3443182 1.9260292 3.7086787 1.1968261 1.3988595
## New Hampshire 3.5619825 4.8650686 4.2430411 2.4949861 5.1270892 4.1126287
## New Jersey 2.6980230 4.1791832 2.1755787 2.7398478 2.7463023 2.3229870
## New Mexico 1.5993970 2.0580889 1.0376848 2.3183196 1.8010201 1.5467439
## New York 2.0723680 3.2903769 1.0725219 2.7478626 1.6787069 1.7363385
## North Carolina 1.6043662 3.2403071 3.1478947 2.0717938 4.2802569 3.8649275
## North Dakota 4.0614988 5.2110254 4.9319844 2.8756492 5.8660699 4.8014019
## Ohio 2.2698519 3.5903348 2.3585705 1.9617104 3.0133425 2.1188236
## Oklahoma 1.9570874 3.3416664 2.2648377 1.4224574 3.1488712 2.2263966
## Oregon 2.3705678 2.6990696 2.0008664 1.8477626 2.6574019 1.5331980
## Pennsylvania 2.5161340 4.1239537 2.9188907 1.9739986 3.7144562 2.8541709
## Rhode Island 3.3951297 5.0629572 3.0570151 3.0883430 3.8883995 3.4810739
## South Carolina 0.9157968 2.5640542 2.7992041 1.7074195 3.7546959 3.2131137
## South Dakota 3.0835587 4.2467198 4.1020099 1.8724822 5.0529153 3.9667318
## Tennessee 0.8407489 2.3362541 2.2989846 1.4254486 3.0119267 2.1972111
## Texas 1.6463225 3.1527905 1.6448574 2.3505545 2.1698156 1.7947199
## Utah 3.0906007 3.9480881 2.5244431 2.6049855 3.0701663 2.2461228
## Vermont 3.9791527 4.8707876 5.1003665 2.7442984 6.0323504 4.8924735
## Virginia 1.4859733 3.0492081 2.3106550 0.9971035 3.2159723 2.2622539
## Washington 2.6481824 3.2715253 2.1399117 2.1313402 2.7746720 1.7897920
## West Virginia 3.1243471 4.5004558 4.4974190 1.9951691 5.4883565 4.4210375
## Wisconsin 3.5047330 4.8711543 3.9425867 2.6102451 4.7354960 3.7846917
## Wyoming 1.8291027 3.4993456 2.6923028 0.9912639 3.7242766 2.8211492
## Connecticut Delaware Florida Georgia Hawaii Idaho
## Alaska
## Arizona
## Arkansas
## California
## Colorado
## Connecticut
## Delaware 1.7568475
## Florida 4.4700701 3.0614170
## Georgia 3.9738227 2.9838715 2.1812958
## Hawaii 1.3843291 2.4748807 4.3596338 3.8105218
## Idaho 1.6354214 2.0382540 4.6999827 3.8005715 2.3658101
## Illinois 2.7400560 1.5584719 1.7711863 2.3135778 2.7329756 3.2728945
## Indiana 1.6147898 1.6973340 3.6150778 2.6924143 1.5460727 1.4923351
## Iowa 1.5470089 2.6068606 5.2682765 4.2517889 2.1564575 0.8584962
## Kansas 1.2280424 1.5510864 3.8424558 3.0071474 1.4648766 1.2103118
## Kentucky 2.3346386 2.2514939 3.9474983 2.4408198 2.5203345 1.6565236
## Louisiana 3.5329409 2.3266996 1.7529677 0.8592544 3.5687157 3.5283772
## Maine 1.8792141 2.6560808 5.3946798 4.3334217 2.7160558 0.8486112
## Maryland 3.4968269 1.9624834 1.4355204 1.8388691 3.6148670 3.4014584
## Massachusetts 0.9468199 1.4382527 3.7753087 3.6706708 1.3276676 2.2201020
## Michigan 3.7037870 2.5165292 1.3357020 1.9185489 3.4123472 3.7775301
## Minnesota 0.9843793 2.1652930 4.7635252 3.9621842 1.4673850 1.0124936
## Mississippi 4.1762631 3.0510628 3.0886673 1.5828594 4.4777223 3.6002946
## Missouri 2.4383227 1.6723281 2.5182466 2.1021909 2.1832480 2.4697182
## Montana 1.8584328 2.0306850 4.2696476 3.0967288 2.2488801 0.8286936
## Nebraska 1.2116949 1.8113430 4.3082894 3.4295510 1.6628657 0.7515014
## Nevada 4.5868149 3.5920897 1.9500388 2.9023041 4.0281974 4.7300228
## New Hampshire 1.6169000 2.6744233 5.3778074 4.3427351 2.3112009 0.9249563
## New Jersey 1.6108823 1.5808719 3.1900596 3.1989350 1.5050500 2.7425260
## New Mexico 3.6233659 2.2271650 1.2965798 1.9015384 3.5506088 3.5883476
## New York 3.0239174 1.8992106 1.5730970 2.3634498 2.9055803 3.5910319
## North Carolina 4.1894604 2.7475286 2.9994188 2.3351307 4.7330517 3.5929592
## North Dakota 2.5099838 3.3615239 6.0356613 4.8596758 3.1974906 1.4144557
## Ohio 1.4443671 1.5838515 3.3897305 2.8043208 1.1494313 1.9647327
## Oklahoma 1.4510623 1.1802929 3.3553471 2.7121515 1.6585736 1.5168111
## Oregon 2.1756954 1.7742778 3.3399718 2.9998878 2.0031861 1.9757247
## Pennsylvania 0.8721491 1.5894850 3.9389869 3.1817981 1.2119256 1.5171866
## Rhode Island 1.0756115 1.6230495 4.2314871 4.1832075 2.0590981 2.4592705
## South Carolina 4.0127954 2.7039667 2.5295912 1.3970074 4.2531214 3.4549959
## South Dakota 2.2397424 2.6722813 5.1015141 3.8729745 2.8044891 0.8070290
## Tennessee 3.2302375 2.3195070 2.3992285 1.0122252 3.0747375 2.9234395
## Texas 2.8734475 2.0031365 1.8537984 1.7575559 2.5901696 3.3172180
## Utah 1.2825907 1.8080931 3.9274528 3.7183994 1.0709720 2.0268663
## Vermont 3.2066152 3.7144653 6.0766416 4.7091538 3.7208347 1.7797462
## Virginia 1.9277004 1.4088230 3.1515587 2.2249559 2.0479238 1.6999289
## Washington 1.6963486 1.6350170 3.5570666 3.3016469 1.5452901 1.8861921
## West Virginia 2.7117590 3.0381601 5.3004067 3.8545331 3.2831874 1.4398440
## Wisconsin 1.0354597 2.4410507 5.1085370 4.2281611 1.6666970 1.2105401
## Wyoming 1.6218573 1.2586225 3.6325811 2.7329062 2.1883414 1.1687896
## Illinois Indiana Iowa Kansas Kentucky Louisiana
## Alaska
## Arizona
## Arkansas
## California
## Colorado
## Connecticut
## Delaware
## Florida
## Georgia
## Hawaii
## Idaho
## Illinois
## Indiana 2.2027081
## Iowa 3.7380070 1.7786548
## Kansas 2.3228505 0.4287712 1.4699265
## Kentucky 2.8478883 1.1790552 1.9426473 1.3020180
## Louisiana 1.6535178 2.4957547 4.0359614 2.7284126 2.4221964
## Maine 3.9342034 2.1029158 0.6457158 1.7913753 1.9925855 4.0901924
## Maryland 1.3429997 2.5430878 4.0642448 2.7400943 2.8229479 1.2739137
## Massachusetts 2.0080982 1.6615695 2.3510287 1.4343401 2.6284451 3.1524549
## Michigan 1.3959090 2.6118471 4.3248636 2.9020920 3.1163494 1.6677999
## Minnesota 3.1558788 1.3184866 0.7644384 0.9745872 1.9333640 3.6905974
## Mississippi 3.0869477 3.0859068 4.1603272 3.2683740 2.3898884 1.6268879
## Missouri 1.3552973 1.2203931 2.9398546 1.5192717 1.9677184 1.8362172
## Montana 2.9659043 1.0033431 1.2403561 0.9170466 0.8523702 2.9444756
## Nebraska 2.7962196 0.8570429 0.9821819 0.5279092 1.4219429 3.1706333
## Nevada 2.3891753 3.5278633 5.2227312 3.8391728 4.1644286 2.8410670
## New Hampshire 3.8490624 1.9278736 0.2058539 1.6084091 2.0093558 4.1168122
## New Jersey 1.4562775 1.7638332 2.9122979 1.7071034 2.6914828 2.6826380
## New Mexico 1.3393276 2.5909993 4.2131394 2.8356373 3.0007332 1.4911656
## New York 0.3502188 2.4628527 4.0411586 2.6096016 3.1213366 1.7495096
## North Carolina 3.0124311 3.3437548 4.2973973 3.4387635 2.8798080 1.9868618
## North Dakota 4.6139615 2.6587932 1.0534375 2.3970805 2.4482563 4.6977846
## Ohio 1.8124981 0.6976320 2.1610242 0.7817000 1.7726720 2.4996969
## Oklahoma 1.8439860 0.5303259 1.9391446 0.5198728 1.4623483 2.3535566
## Oregon 2.0743434 1.1780815 2.4662295 1.3426890 2.1388677 2.7490592
## Pennsylvania 2.3134187 0.8412900 1.5708895 0.5456840 1.5944097 2.8440845
## Rhode Island 2.5057761 2.3335609 2.5453686 2.0087021 3.0457816 3.5648047
## South Carolina 2.6163680 2.8469842 4.1015324 3.0609333 2.4166385 1.3151908
## South Dakota 3.8004708 1.8411735 0.9886706 1.6701106 1.5114990 3.7457555
## Tennessee 1.9478353 1.8100316 3.4176329 2.1533060 1.7489942 1.1298534
## Texas 0.8241352 2.0035762 3.6962443 2.2378289 2.5297839 1.3325285
## Utah 2.2771632 1.4019666 2.1682069 1.2751603 2.5461745 3.3440990
## Vermont 4.8624402 2.8667983 1.7298425 2.7298377 2.3888326 4.6795933
## Virginia 1.8624960 0.6127246 2.1704984 0.8351949 1.0918624 1.9554079
## Washington 2.0612962 1.1405746 2.2502832 1.1579118 2.2630242 2.9705622
## West Virginia 4.1148082 2.2478563 1.5256890 2.1244674 1.5236299 3.7947215
## Wisconsin 3.4790637 1.6806129 0.6318069 1.3242947 2.0950212 3.9559184
## Wyoming 2.2643574 0.8898783 1.7194683 0.7588728 1.0694408 2.3837077
## Maine Maryland Massachusetts Michigan Minnesota
## Alaska
## Arizona
## Arkansas
## California
## Colorado
## Connecticut
## Delaware
## Florida
## Georgia
## Hawaii
## Idaho
## Illinois
## Indiana
## Iowa
## Kansas
## Kentucky
## Louisiana
## Maine
## Maryland 4.1259083
## Massachusetts 2.6920282 2.9743193
## Michigan 4.5333420 1.0800988 3.0576915
## Minnesota 1.2980362 3.6448929 1.6587245 3.7995101
## Mississippi 4.0014591 2.2992240 4.1217248 2.9722824 4.1067600
## Missouri 3.2055955 1.5705755 1.9810531 1.4068840 2.4088795
## Montana 1.3271199 3.0249456 2.2919046 3.3348908 1.2662635
## Nebraska 1.3218907 3.1309065 1.6863806 3.3478988 0.6083415
## Nevada 5.5153139 2.2551337 3.8556049 1.2609417 4.6391114
## New Hampshire 0.4995971 4.1663744 2.4573524 4.4646172 0.9279247
## New Jersey 3.2532459 2.6263456 0.7977642 2.5678440 2.2254151
## New Mexico 4.3460538 0.5353893 3.0274701 0.5782474 3.7377675
## New York 4.2595904 1.4362170 2.2479437 1.2897453 3.4391596
## North Carolina 4.0631653 2.0542355 4.0773401 3.0232021 4.2219622
## North Dakota 0.7305609 4.7423030 3.3446903 5.1171939 1.8065731
## Ohio 2.5455752 2.5061694 1.1567960 2.4459855 1.5216293
## Oklahoma 2.1929825 2.2492942 1.3383233 2.4336743 1.4198434
## Oregon 2.7813372 2.2466329 1.8709252 2.1626274 1.9270100
## Pennsylvania 1.9197571 2.9585539 1.1337883 3.1048542 1.0106613
## Rhode Island 2.7331079 3.4379146 0.9440940 3.7320501 2.0310592
## South Carolina 4.0015575 1.6165582 3.8310425 2.3233363 3.9484630
## South Dakota 0.7812991 3.7991896 2.8925136 4.1744724 1.4990317
## Tennessee 3.5420469 1.5202431 2.9678843 1.5970196 3.1023238
## Texas 3.9386296 1.5431868 2.2593978 1.2888621 3.1438264
## Utah 2.6218087 3.0338001 0.9015809 2.9441421 1.4177147
## Vermont 1.4253680 4.7430576 3.9277625 5.1250778 2.4019924
## Virginia 2.3474650 2.0124420 1.8503795 2.2439957 1.7932233
## Washington 2.6292546 2.5434911 1.3472994 2.4715215 1.5955418
## West Virginia 1.1818120 4.0251562 3.3782752 4.4668346 2.0791705
## Wisconsin 1.1485830 4.0091486 1.8882704 4.2034334 0.4940832
## Wyoming 1.7665064 2.4041294 1.8201580 2.8324573 1.4845967
## Mississippi Missouri Montana Nebraska Nevada
## Alaska
## Arizona
## Arkansas
## California
## Colorado
## Connecticut
## Delaware
## Florida
## Georgia
## Hawaii
## Idaho
## Illinois
## Indiana
## Iowa
## Kansas
## Kentucky
## Louisiana
## Maine
## Maryland
## Massachusetts
## Michigan
## Minnesota
## Mississippi
## Missouri 2.8692946
## Montana 3.0015255 2.0313649
## Nebraska 3.5269565 1.9651798 0.7389936
## Nevada 4.1064793 2.3489003 4.3243112 4.2628916
## New Hampshire 4.1895936 3.0885710 1.3329504 1.1300720 5.3871427
## New Jersey 3.8894324 1.7079555 2.5912431 2.1246377 3.3464214
## New Mexico 2.6557350 1.4579057 3.1915871 3.2494088 1.7234839
## New York 3.2655822 1.5284764 3.2662661 3.0925340 2.1674148
## North Carolina 1.1826891 3.0224849 3.2209267 3.6500186 4.1773437
## North Dakota 4.4753078 3.7811273 1.8291157 1.9038740 6.0519445
## Ohio 3.4148987 1.1327425 1.6436336 1.2654510 3.2930712
## Oklahoma 3.0466140 1.0927654 1.2225315 0.9674809 3.4108696
## Oregon 3.5033774 0.9974171 1.8044622 1.5727910 2.8581280
## Pennsylvania 3.4971746 1.7793568 1.3246445 0.8483058 4.0392694
## Rhode Island 4.3875001 2.7475003 2.6888576 2.1303973 4.6185330
## South Carolina 0.7865674 2.3846001 2.9024302 3.3517226 3.4427701
## South Dakota 3.5355186 2.8862448 0.8857149 1.2591419 5.1416772
## Tennessee 1.8269569 1.2413874 2.2494023 2.5526834 2.6666268
## Texas 2.8431727 1.1654171 2.8298991 2.7568751 2.2765693
## Utah 4.2571173 1.7478909 2.0956369 1.4573012 3.5868975
## Vermont 4.2046660 3.8803394 1.9261350 2.2952287 6.0437845
## Virginia 2.5383053 0.9787310 1.1556682 1.2472262 3.3001850
## Washington 3.8140404 1.2502752 1.8442691 1.3859985 3.1570805
## West Virginia 3.3281129 3.2538044 1.2758193 1.8117833 5.4963193
## Wisconsin 4.2987974 2.8171535 1.4916365 0.9719877 5.0751736
## Wyoming 2.6813279 1.6073860 0.8150071 0.9268202 3.9202716
## New Hampshire New Jersey New Mexico New York North Carolina
## Alaska
## Arizona
## Arkansas
## California
## Colorado
## Connecticut
## Delaware
## Florida
## Georgia
## Hawaii
## Idaho
## Illinois
## Indiana
## Iowa
## Kansas
## Kentucky
## Louisiana
## Maine
## Maryland
## Massachusetts
## Michigan
## Minnesota
## Mississippi
## Missouri
## Montana
## Nebraska
## Nevada
## New Hampshire
## New Jersey 3.0269198
## New Mexico 4.3360809 2.6208087
## New York 4.1586415 1.6344744 1.3324096
## North Carolina 4.3157112 3.9418824 2.5348334 3.2163998
## North Dakota 0.9231894 3.9166205 4.9450519 4.9325292 4.5836787
## Ohio 2.3095495 1.1099823 2.4960904 2.0434995 3.6205693
## Oklahoma 2.0697098 1.4711183 2.3426252 2.1367108 3.1366639
## Oregon 2.6377191 1.9738854 2.1553130 2.2727718 3.5095191
## Pennsylvania 1.6822035 1.4216058 3.0619915 2.5949374 3.6803956
## Rhode Island 2.5813199 1.4668378 3.6032966 2.7682543 4.2185789
## South Carolina 4.1596914 3.5826726 1.9596343 2.7755634 1.0476313
## South Dakota 0.9874611 3.3318222 3.9969513 4.1124693 3.6955387
## Tennessee 3.5298430 2.6339707 1.5528304 2.0847931 2.3374653
## Texas 3.8178258 1.6226525 1.4418241 0.8457697 3.0857436
## Utah 2.3304873 1.3141843 2.9843796 2.4826984 4.2680823
## Vermont 1.6716127 4.4005416 4.9416825 5.1704762 4.3880034
## Virginia 2.2878085 1.8255601 2.1341562 2.1439207 2.7517523
## Washington 2.4214987 1.5759539 2.4796057 2.2747965 3.8055684
## West Virginia 1.4648924 3.7402121 4.2681325 4.4279608 3.5978058
## Wisconsin 0.7155628 2.4671212 4.1327758 3.7687073 4.4429456
## Wyoming 1.7950754 2.0372127 2.6286722 2.5890441 2.7501141
## North Dakota Ohio Oklahoma Oregon Pennsylvania
## Alaska
## Arizona
## Arkansas
## California
## Colorado
## Connecticut
## Delaware
## Florida
## Georgia
## Hawaii
## Idaho
## Illinois
## Indiana
## Iowa
## Kansas
## Kentucky
## Louisiana
## Maine
## Maryland
## Massachusetts
## Michigan
## Minnesota
## Mississippi
## Missouri
## Montana
## Nebraska
## Nevada
## New Hampshire
## New Jersey
## New Mexico
## New York
## North Carolina
## North Dakota
## Ohio 3.1448279
## Oklahoma 2.8246690 0.6483903
## Oregon 3.2862071 1.2407607 1.0734082
## Pennsylvania 2.5555137 0.7781298 0.8180221 1.7293732
## Rhode Island 3.4042300 1.9659747 1.9746699 2.6621371 1.6369255
## South Carolina 4.5104172 3.1289884 2.7470931 3.0134453 3.3429642
## South Dakota 1.0324944 2.4394250 2.0340486 2.4988870 1.9790714
## Tennessee 4.0623149 2.0167804 1.8500296 2.0306758 2.4343114
## Texas 4.5749422 1.6711510 1.8312655 2.1053000 2.2460705
## Utah 3.1738212 1.0154223 1.2372916 1.2825152 1.2529078
## Vermont 0.9824857 3.4825859 3.1010306 3.4262789 3.0270572
## Virginia 2.9443461 0.9774388 0.5646254 1.2664430 1.1769236
## Washington 3.1725909 0.9725013 0.9586525 0.5935343 1.3993323
## West Virginia 1.2716808 2.8650371 2.4631736 3.0349855 2.3799278
## Wisconsin 1.6216339 1.8649801 1.7916829 2.4088700 1.2204658
## Wyoming 2.4170757 1.3086480 0.7366465 1.6013015 1.0684605
## Rhode Island South Carolina South Dakota Tennessee Texas
## Alaska
## Arizona
## Arkansas
## California
## Colorado
## Connecticut
## Delaware
## Florida
## Georgia
## Hawaii
## Idaho
## Illinois
## Indiana
## Iowa
## Kansas
## Kentucky
## Louisiana
## Maine
## Maryland
## Massachusetts
## Michigan
## Minnesota
## Mississippi
## Missouri
## Montana
## Nebraska
## Nevada
## New Hampshire
## New Jersey
## New Mexico
## New York
## North Carolina
## North Dakota
## Ohio
## Oklahoma
## Oregon
## Pennsylvania
## Rhode Island
## South Carolina 4.1861320
## South Dakota 3.1262712 3.5215978
## Tennessee 3.5743861 1.4375120 3.0589938
## Texas 2.8757996 2.4532276 3.7101039 1.4712840
## Utah 1.7565845 3.8912317 2.6823382 2.8678113 2.4039834
## Vermont 4.1104165 4.2668977 1.0856574 3.9356721 4.7444455
## Virginia 2.4330133 2.2636538 2.0316897 1.3514491 1.6921625
## Washington 2.1743525 3.3802314 2.5083824 2.3809584 2.1635337
## West Virginia 3.5400858 3.4651680 0.7108812 3.1707450 3.9586581
## Wisconsin 2.0779526 4.2190973 1.5437375 3.4257189 3.4539515
## Wyoming 2.1726807 2.5059056 1.5644785 1.9298669 2.2564704
## Utah Vermont Virginia Washington West Virginia Wisconsin
## Alaska
## Arizona
## Arkansas
## California
## Colorado
## Connecticut
## Delaware
## Florida
## Georgia
## Hawaii
## Idaho
## Illinois
## Indiana
## Iowa
## Kansas
## Kentucky
## Louisiana
## Maine
## Maryland
## Massachusetts
## Michigan
## Minnesota
## Mississippi
## Missouri
## Montana
## Nebraska
## Nevada
## New Hampshire
## New Jersey
## New Mexico
## New York
## North Carolina
## North Dakota
## Ohio
## Oklahoma
## Oregon
## Pennsylvania
## Rhode Island
## South Carolina
## South Dakota
## Tennessee
## Texas
## Utah
## Vermont 3.6546040
## Virginia 1.7612066 3.0638337
## Washington 0.6940667 3.4804319 1.3809295
## West Virginia 3.2680139 1.0380554 2.3353210 3.0846553
## Wisconsin 1.8082282 2.3518637 2.1266497 2.0637823 2.0308890
## Wyoming 1.8552036 2.6299335 0.7038309 1.5929546 1.8821600 1.7446366Sekarang kita siap menggunakan fungsi hclust() untuk melakukan klasterisasi menggunakan metode “hierarchical clustering”.
hc <-hclust(d)
hc##
## Call:
## hclust(d = d)
##
## Cluster method : complete
## Distance : euclidean
## Number of objects: 50Kita dapat melihat bagaimana metode ini mengelompokkan data secara bersama-sama dengan menggunakan “dendrogram”. Ada beberapa cara untuk melihat berapa banyak kelompok (klaster) yang ada dalam dataset. Dari “dendrogram”, kita dapat mengetahui bahwa pada dasarnya terdapat empat kelompok dalam dataset tersebut. Oleh karena itu, kita menggabungkan keempat kelompok tersebut menjadi satu.
sub_grp <-cutree(hc,k=4)
sub_grp## Alabama Alaska Arizona Arkansas California
## 1 1 2 3 2
## Colorado Connecticut Delaware Florida Georgia
## 2 3 3 2 1
## Hawaii Idaho Illinois Indiana Iowa
## 3 4 2 3 4
## Kansas Kentucky Louisiana Maine Maryland
## 3 3 1 4 2
## Massachusetts Michigan Minnesota Mississippi Missouri
## 3 2 3 1 3
## Montana Nebraska Nevada New Hampshire New Jersey
## 4 4 2 4 3
## New Mexico New York North Carolina North Dakota Ohio
## 2 2 1 4 3
## Oklahoma Oregon Pennsylvania Rhode Island South Carolina
## 3 3 3 3 1
## South Dakota Tennessee Texas Utah Vermont
## 4 1 2 3 4
## Virginia Washington West Virginia Wisconsin Wyoming
## 3 3 4 3 3Kemudian kita menggunakan paket factoextra untuk menggambar plot dari klaster yang terbentuk.
library(factoextra)## Loading required package: ggplot2## Welcome! Want to learn more? See two factoextra-related books at https://goo.gl/ve3WBaSekarang kita akan menggambar hasilnya. Kita akan menggunakan fungsi fviz_cluster() untuk melakukan itu.
fviz_cluster(list(data =df,cluster=sub_grp))
Klaster dalam data “USArrests” melalui metode klasterisasi hirarkis. Setiap klaster memiliki warna yang unik. Sumbu x mewakili komponen utama pertama (first principal component) dan sumbu y mewakili komponen utama kedua (second principal component) yang dihitung melalui PCA (Principal Component Analysis).
Baik, cukup sekian pembahasan tentang ruang pengantar contoh Data Science.