Indeks Pembangunan Manusia di Indonesia Tahun 2021
Pramita Dea Z.
2024-11-06
1 PENDAHULUAN
1.1 Latar Belakang
Indeks Pembangunan Manusia (IPM) merupakan indikator yang digunakan untuk mengukur kualitas hidup, kemajuan pembangunan, dan kesejahteraan masyarakat pada suatu negara. Konsep IPM pertama kali diperkenalkan oleh United Nations Development Program (UNDP) pada tahun 1990. IPM diukur berdasarkan pada aspek sosial, ekonomi, serta mencakup tiga dimensi yaitu umur panjang dan hidup sehat, pengetahuan, dan standar hidup layak. Ketiga dimensi tersebut dapat diukur melalui beberapa indikator seperti persentase penduduk miskin, rata-rata lama sekolah, pengeluaran per kapita, umur harapan hidup, persentase rumah tangga yang memiliki akses terhadap sanitasi layak, persentase rumah tangga yang memiliki akses terhadap air minum layak, tingkat pengangguran terbuka, tingkat partisipasi angkatan kerja, dan PDRB atas dasar harga konstan menurut pengeluaran.
Di Indonesia IPM mulai dihitung pada tahun 1996. Tujuannya adalah agar dapat mengetahui perkembangan IPM di seluruh wilayah Indonesia. IPM di Indonesia terus mengalami peningkatan setiap tahunnya. Contohnya IPM pada tahun 2021 mencapai 72,29, meningkat sebesar 0,49 persen dari tahun sebelumnya. Meskipun IPM terus mengalami peningkatan, masih terdapat tantangan yang harus dihadapi seperti kesenjangan dalam program pemerintah di berbagai wilayah. Oleh karena itu, perlu dilakukan analisis mengenai IPM agar mendapatkan strategi dan solusi dalam meningkatkan kualitas hidup masyarakat Indonesia.
1.2 Data
Data yang digunakan berasal dari https://www.kaggle.com/ dengan judul “IPM Indonesia”. Data yang digunakan terdiri dari seluruh kota dan kabupaten di Indonesia yaitu sebanyak 514 dengan 9 variabel dengan rincian:
| Variabel | Pernyataan |
|---|---|
| X1 | Persentase penduduk miskin |
| X2 | Rata-rata lama sekolah |
| X3 | Pengeluaran per kapita |
| X4 | Umur harapan hidup |
| X5 | Persentase rumah tangga yang memiliki akses terhadap sanitasi layak |
| X6 | Persentase rumah tangga yang memiliki akses terhadap air minum layak |
| X7 | Tingkat pengangguran terbuka |
| X8 | Tingkat partisipasi angkatan kerja |
| X9 | PDRB atas dasar harga konstan menurut pengeluaran |
Data Indeks Pembangunan Manusia di Indonesia Tahun 2021
> library(readxl)
> data <- read_excel("C:/Users/prami/OneDrive/Documents/data ipm.xlsx")
> library(DT)
> datatable(data)1.3 Latar Belakang Metode
Data yang digunakan merupakan data indeks pembangunan manusia di Indonesia tahun 2021. Pada penelitian ini terdapat 9 faktor yang mempengaruhi IPM di Indonesia yaitu persentase penduduk miskin, rata-rata lama sekolah, pengeluaran per kapita, umur harapan hidup, persentase rumah tangga yang memiliki akses terhadap sanitasi layak, persentase rumah tangga yang memiliki akses terhadap air minum layak, tingkat pengangguran terbuka, tingkat partisipasi angkatan kerja, dan PDRB atas dasar harga konstan menurut pengeluaran. Hasil IPM diperoleh dari seluruh kota dan kabupaten yang ada di Indonesia.
Metode analisis faktor dan PCA dilakukan karena pada penelitian ini digunakan banyak variabel sehingga kesulitan dalam melakukan analisis dan menentukan interpretasi. Dengan menggunakan metode analisis tersebut kita dapat mereduksi variabel menjadi beberapa komponen (faktor) yang dapat mewakili seluruh variabel yang digunakan.
1.4 Tinjauan Pustaka Metode
Analisis faktor dan Principal Component Analysis (PCA) atau analisis komponen utama adalah metode statistik yang digunakan untuk menganalisis variabel dan menjelaskan faktornya. Tujuan analisis tersebut adalah untuk meringkas informasi dari sejumlah variabel asli menjadi sebagian kecil faktor.
Principal Component Analysis (PCA) atau analisis komponen utama adalah metode statistik yang dimanfaatkan untuk mentransformasi sebagian besar variabel asli yang tidak bebas menjadi variabel baru yang lebih sedikit dan bersifat bebas. PCA bertujuan untuk mereduksi dimensi data tanpa mengurangi informasi dari data tersebut. Langkah-langkah yang dilakukan dalam PCA adalah dengan menghitung mean, matriks kovarian, serta nilai eigen dari matriks kovarian tersebut.
Analisis faktor adalah metode statistik yang digunakan untuk mereduksi variabel menjadi faktor (variabel baru) yang jumlahnya lebih sedikit. Variabel baru tersebut menggambarkan informasi variabel asli sebelumnya. Tahapan analisis faktor yaitu penentuan variabel yang akan dianalisis, Bartletts Test of Sphericity, Kaiser-Meyer-Olkin (KMO), Measure of Sampling Adequacy (MSA), ekstraksi faktor, serta penentuan jumlah dan skor faktor (Subagja dan Disan, 2020).
Uji Kaiser-Meyer-Olkin (KMO) digunakan untuk mengukur kecocokan data untuk analisis faktor. Suatu data dikatakan sudah valid dilanjutkan analisis faktor jika koefisien korelasi sampel teramati dengan koefisien korelasi parsial memiliki selang nilai antara 0.5 hingga 1 (Subagja dan Disan, 2020). Jika nilai KMO sudah berada pada selang tersebut, maka penelitian dapat dianalisis lebih lanjut.
Bartletts Test of Sphericity dilakukan untuk menyatakan bahwa tidak ada korelasi antar variabel (homoskedastisitas). Jika pada Bartletts Test of Sphericity diperoleh nilai signifikan kurang dari 0.05, maka dapat disimpulkan bahwa korelasi antar variabel bebasnya telah signifikan dan dianggap layak untuk dianalisis lebih lanjut.
Measure of Sampling Adequacy (MSA) digunakan untuk menentukan apakah variabel sudah layak untuk dianalisis lebih lanjut. Jika nilai MSA < 0,5 maka indikator tersebut akan tereliminasi dan tidak dapat diikutsertakan dalam analisis komponen utama.
1.5 Tujuan
- Mengetahui faktor-faktor yang mempengaruhi indeks pembangunan manusia di Indonesia tahun 2021.
- Mengetahui faktor yang paling berpengaruh terhadap indeks pembangunan manusia di Indonesia tahun 2021.
2 Source Code
2.1 Library
> library(readxl) #untuk memuat package readxl, untuk membaca file Excel.
> library(DT) #untuk memuat package DT, yang menyediakan alat untuk membuat tabel interaktif.
> library(readr) #untuk memuat package readr. Untuk membaca data dari berbagai jenis file teks ke dalam R, seperti file CSV, TSV, dan file teks lainnya.
> library(dplyr) #untuk memuat package dplyr di R. Untuk manipulasi dan analisis data.
> library(corrplot) #untuk memuat package corrplot. Untuk membuuat visualisasi matriks korelasi.
> library(REdaS) #untuk memuat package REdaS, berisi fungsi untuk melakukan uji KMO dan MSA.
> library(psych) #untuk memuat package psych, untuk melakukan berbagai jenis analisis faktor, PCA, dan rotasi faktor.3 Hasil dan Pembahasan
3.1 PCA
- Statistika Deskriptif
> library(readr)
> Data <- read_delim("C:/Users/prami/OneDrive/Documents/dataa ipm.csv", delim = ";", trim_ws = TRUE)
> # untuk membaca file yang dipisahkan oleh delimiter tertentu, dalam hal ini tanda ;. Argumen trim_ws = TRUE memastikan bahwa spasi kosong dihilangkan dari data.
> View(Data)
> summary(Data)
Provinsi Kab/Kota X1 X2
Length:514 Length:514 Min. : 2.38 Min. : 1.420
Class :character Class :character 1st Qu.: 7.15 1st Qu.: 7.510
Mode :character Mode :character Median :10.46 Median : 8.305
Mean :12.27 Mean : 8.437
3rd Qu.:14.89 3rd Qu.: 9.338
Max. :41.66 Max. :12.830
X3 X4 X5 X6
Min. : 3976 Min. :55.43 Min. : 0.00 Min. : 0.00
1st Qu.: 8574 1st Qu.:67.39 1st Qu.:70.22 1st Qu.: 79.04
Median :10196 Median :69.97 Median :81.80 Median : 89.80
Mean :10325 Mean :69.66 Mean :77.20 Mean : 85.14
3rd Qu.:11719 3rd Qu.:72.04 3rd Qu.:89.88 3rd Qu.: 96.40
Max. :23888 Max. :77.73 Max. :99.97 Max. :100.00
X7 X8 X9
Min. : 0.000 Min. :56.39 Min. : 147485
1st Qu.: 3.180 1st Qu.:65.07 1st Qu.: 3654292
Median : 4.565 Median :68.95 Median : 8814926
Mean : 5.059 Mean :69.46 Mean : 21964077
3rd Qu.: 6.530 3rd Qu.:72.34 3rd Qu.: 19735101
Max. :13.370 Max. :97.93 Max. :460081046 Interpretasi:
Variabel Provinsi: Terdapat 514 observasi atau data untuk setiap provinsi (Length:514). Data ini termasuk variabel karakter, sehingga tidak ada perhitungan seperti mean, median, atau rentang.
Variabel Kab/Kota: Terdapat 514 observasi atau data untuk setiap kabupaten dan kota (Length:514). Data ini termasuk variabel karakter, sehingga tidak ada perhitungan seperti mean, median, atau rentang.
Variabel X1-X9:
Min: Nilai minimum yang terdapat pada setiap variabel. Contohnya pada X1 nilai minimumnya adalah 2.38 dan pada X2 nilai minimumnya adalah 1.42.
1st Quartile (1st Qu.) adalah nilai kuartil pertama, yang berarti 25% dari data berada di bawah nilai ini. Contohnya untuk X1, kuartil pertama adalah 7.15, menunjukkan bahwa 25% dari data X1 memiliki nilai lebih rendah dari ini.
Median merupakan nilai tengah dari data, artinya 50% dari data berada di atas nilai ini, dan 50% di bawah. Misalnya, median X1 adalah 10.46, artinya separuh atau 50% dari data X1 memiliki nilai di bawah 10.46 dan separuh lainnya di atas.
Mean adalah rata-rata dari data. Contohnya, nilai mean X1 adalah 12.27
3st Quartile (3st Qu.) adalah nilai kuartil ketiga, yang menunjukkan bahwa 75% dari data berada di bawah nilai ini. Contohnya X1 memiliki kuartil ketiga sebesar 14.89.
Max adalah nilai maksimum yang terdapat dalam data untuk setiap variabel. Contohnya X1 memiliki nilai maksimum 41.66, artinya merupakan nilai tertinggi untuk variabel ini.
Eksplorasi Korelasi
> library(dplyr)
> library(corrplot)
> Data_numeric <- Data %>% select_if(is.numeric) #Menyeleksi hanya kolom numerik dari dataset menggunakan pipe %>% dari dplyr.
> kor<- cor(Data_numeric) #Menghitung matriks korelasi antar variabel numerik.
> corrplot(kor, method = "number") #Menampilkan korelasi antar variabel dalam bentuk visual menggunakan angka pada setiap sel matriks.
Keterangan:
Warna biru tua menunjukkan korelasi positif yang kuat.
Warna merah tua menunjukkan korelasi negatif yang kuat.
Warna putih menunjukkan korelasi lemah atau tidak ada korelasi.
Interpretasi:
X1 dan X3 memiliki korelasi negatif yang sangat kuat (-0.64). Artinya ketika nilai X1 meningkat, X3 cenderung menurun dan sebaliknya.
X2 dan X3 memiliki korelasi positif yang sangat kuat (0.67). Ini menunjukkan bahwa ketika nilai X2 meningkat, X3 juga cenderung meningkat. Begitu pula sebaliknya.
Menampilkan Kumulatif Nilai Eigen dan Vektor Eigen Dengan Matriks Kovarians
Dekomposisi Eigen:
> sc <- scale(Data_numeric) #Melakukan standardisasi data (mean = 0, standar deviasi = 1) agar PCA lebih akurat, terutama jika skala antar variabel sangat berbeda.
> sc
X1 X2 X3 X4 X5
[1,] 0.8991976456 0.639783209 -1.1691642845 -1.269573591 -0.3036145378
[2,] 1.0842164062 0.149239007 -0.5700057972 -0.645926772 -0.4112365745
[3,] 0.1215825646 0.271875057 -0.7893537446 -1.524833684 -0.7884518132
[4,] 0.1524190247 0.756287457 -0.8445587650 -0.416772825 -0.7798420503
[5,] 0.2918534530 -0.138955712 -0.6432444575 -0.265937315 -0.5624455361
[6,] 0.4004514212 0.872791705 0.1675332743 -0.231129121 0.7198710313
[7,] 0.8764054794 0.682705827 -0.2693224532 -0.483488531 0.6671362333
[8,] 0.2382248267 1.160986423 -0.2505527463 0.038634387 0.5487519929
[9,] 0.9809813006 0.345456688 -0.1710575170 -0.785159550 -1.2431549183
[10,] 0.1309675742 0.523278961 -0.5365147516 0.465034770 0.2522532818
[11,] 0.6913867187 0.124711797 -0.7816250418 -0.245632535 0.1489361265
[12,] 0.5452487121 0.143107204 -0.6980814444 -1.333388614 -0.6184089952
[13,] 0.9876848789 -0.022451464 -0.5405631197 -1.197056519 -1.5913122071
[14,] 0.1430340151 0.290270465 -0.7205314860 -0.007776539 0.5514425438
[15,] 0.7986439713 0.155370809 -0.7481339961 -0.120903171 -0.1260381773
[16,] 0.1282861429 0.167634415 -0.2424560100 -0.715543161 0.2366480865
[17,] 0.9233305274 0.958636940 0.2919285868 -0.115101805 0.5105461699
[18,] 0.9756184380 0.553937974 -0.0128031254 0.151761019 -0.1561723475
[19,] -0.6251960563 2.693937054 2.4165858024 0.540452525 1.2203135020
[20,] 0.4084957151 1.682189638 0.3876172887 0.261986969 0.8097354319
[21,] -0.1760563112 1.645398823 0.6411923487 -0.065790196 0.7860585839
[22,] -0.1492419981 1.639267020 0.3920336903 0.575260720 0.8554747975
[23,] 0.7208824632 -0.249328157 -1.0819403524 -1.620556219 -0.4031649218
[24,] 0.6096030637 -1.714828960 -1.2254734052 0.035733704 -3.0818774155
[25,] -0.3731415128 0.118579994 -0.2038124957 -2.032453188 -2.2316633255
[26,] -0.4656508931 0.523278961 0.3603828120 -1.359494760 -1.6569616495
[27,] 0.0532060661 0.247347847 -0.0687442127 -0.701039747 -1.0569687948
[28,] -0.3423050527 0.952505138 0.5098044003 -0.260135949 0.3544942167
[29,] -0.4401772956 1.308149684 0.6989736033 0.183668531 0.6639075722
[30,] -0.4736951871 0.498751751 0.3265237329 0.085045313 0.2312669846
[31,] -0.3919115320 0.222820637 0.2595416415 -0.373262582 0.6396926139
[32,] -0.4643101775 0.719496642 0.3868812218 0.496942282 0.7828299228
[33,] -0.5313459603 0.707233037 0.0659560369 -0.135406586 0.8151165338
[34,] -0.4669916088 0.958636940 0.7681638955 0.557856622 0.3835521666
[35,] -1.1078536927 1.019954965 0.7236318457 0.612969597 1.0314368276
[36,] -0.2886764264 0.136975402 0.3007613900 -0.199221609 0.1914468310
[37,] 0.6230102202 -1.457293254 -1.2085438656 -0.231129121 -3.4472542301
[38,] -0.3516900623 0.780814667 -0.8497112335 -0.042584733 0.7774488209
[39,] -0.3919115320 0.431301923 -0.7621192679 -1.072327155 0.6961941832
[40,] 0.0545467817 0.615255999 -0.6701109007 0.508545013 0.7473146507
[41,] -0.5326866760 0.155370809 0.2547572064 -0.242731852 0.8603177892
[42,] 0.0143253120 -0.224800947 0.0788372083 -0.732947258 0.5831910447
[43,] -0.3154907395 0.578465184 -0.0992909906 -0.706841113 -0.5398449084
[44,] -0.4803987653 0.357720293 -0.5166409442 -0.732947258 -0.9461180970
[45,] -0.5018502158 0.284138662 0.4553354470 -0.245632535 0.4115338961
[46,] -0.3020835830 -0.016319661 0.5576487513 -0.028081319 0.1370977025
[47,] 1.7947957043 -1.021935275 -1.5346215190 -0.030982002 -1.6741811753
[48,] 1.8966900942 -1.070989695 -1.6196372503 -0.167314097 -2.1084360935
[49,] 0.0076217338 1.210040844 0.4472387106 -0.118002488 -2.4146207879
[50,] 0.1510783090 0.621387801 0.3313081680 -1.803299240 0.6386163936
[51,] -0.5031909315 1.749639466 0.7769966987 1.193106173 0.6074060029
[52,] -0.2645435446 1.228436251 0.9621175335 0.375113601 1.0050694286
[53,] -0.5273238133 1.866143714 1.7202664793 1.036469297 0.8344885004
[54,] -0.8665248745 1.535026377 0.2716867460 0.810216033 0.9690160463
[55,] -0.6359217815 1.627003415 0.2356194661 -0.045485416 -1.3922114391
[56,] 0.5599965844 0.112448192 -0.8062832842 0.482438867 -1.7258397529
[57,] 0.3441413636 -0.758267767 -1.4735279632 -1.429111149 -0.3310581572
[58,] -0.5836338709 -0.102164897 -0.3881972636 0.378014284 -0.3122243008
[59,] -0.5715674300 -0.347436997 -0.0404056356 -0.251433901 -1.5913122071
[60,] -0.7337940245 -0.194141934 0.0236321880 -0.956299840 -0.2460367482
[61,] -1.0367957629 0.112448192 0.1071757854 0.134356922 -1.2630649951
[62,] -0.6774839669 -0.341305195 0.2669023109 -0.199221609 -1.1484475260
[63,] -0.7270904462 0.327061280 -0.2439281438 0.833421496 0.0025701566
[64,] -0.6680989573 -0.224800947 -0.2417199430 0.053137802 -1.1575953991
[65,] -0.6426253598 -0.206405539 -0.6936650427 -0.599515846 -1.7188443206
[66,] -0.6372624972 -0.071505884 0.0155354517 -0.477687165 -1.0026196663
[67,] -0.7512233280 0.069525574 0.3677434814 0.543353208 0.2963783168
[68,] -0.6386032128 -0.102164897 -0.4548113215 -0.497991945 -0.3724926413
[69,] -0.9831671367 1.933593541 1.5513391171 1.169900710 0.1575458895
[70,] -1.2271773862 1.596344403 0.6783637291 1.181503441 0.7639960663
[71,] -1.3263903448 1.154854621 -0.0477663049 0.128555556 0.4976315255
[72,] -0.8517770023 1.958120752 0.1579644041 0.917541299 -0.2239742307
[73,] -0.9563528235 1.780298478 1.1063866533 1.404856023 0.6848938694
[74,] -0.8195998265 1.455312945 1.1012341848 1.213410953 0.6612170213
[75,] -1.0582472134 1.369467709 0.9175854838 0.209774677 0.1268736090
[76,] -0.4428587269 0.192161625 -0.0058104895 -0.306546875 0.5799623836
[77,] -0.8169183952 -0.028583266 -0.0238441295 0.174966482 0.3022975289
[78,] -0.8169183952 -0.733740556 -0.1397746721 -0.486389214 -1.4831520602
[79,] -0.3543714936 0.161502612 0.4958191285 0.459233404 0.2113569078
[80,] -0.9509899609 0.872791705 0.5455036468 0.427325893 0.9797782499
[81,] -0.5970410275 0.511015356 0.1962398849 0.340305406 0.6682124537
[82,] -0.2511363880 0.063393772 -0.3381447119 0.151761019 0.6267779695
[83,] -0.7552454750 0.774682864 0.4012345270 0.459233404 0.7069563869
[84,] -0.6828468295 -0.108296699 -0.3340963437 0.212675360 -0.1755443141
[85,] 1.7974771356 -0.365832405 -0.9365671322 -0.544402871 -1.1419902038
[86,] -1.2660581402 2.135943025 1.4850930928 0.798613301 1.0265938359
[87,] -1.1869559165 1.044482175 0.5495520150 0.383815650 1.0669520997
[88,] -0.6117888997 0.075657377 -0.0518146731 0.099548728 -0.5855842740
[89,] -0.4240887078 -0.329041590 0.0203198868 0.473736819 -0.3299819368
[90,] -0.4562658835 -0.243196355 0.5399831448 -0.129605220 -0.0630792858
[91,] -0.2980614360 -0.194141934 -0.1077557604 0.285192432 0.1333309312
[92,] -1.0381364786 0.087920982 -0.5519721572 0.482438867 0.6439974954
[93,] -0.1184055380 -0.929958237 -0.4275768448 -0.962101206 -0.3730307515
[94,] -0.2042113400 -0.267723565 -0.2303109055 -0.431276239 0.2522532818
[95,] -0.7498826124 -0.519127468 0.0814134426 0.105350093 0.3324316991
[96,] -0.8102148169 -0.096033094 0.4950830616 -0.529899457 0.0203277927
[97,] -0.4361551487 1.694453243 0.7048621388 0.885633787 0.8619321197
[98,] -1.1882966322 1.160986423 0.0475543635 0.740599644 -0.1701632123
[99,] 0.0465024878 0.167634415 -0.1048114926 -0.410971459 0.2969164270
[100,] 0.3226899131 -0.850244804 0.1583324376 -0.286242095 -0.4903387715
[101,] 0.0062810181 -0.390359615 0.2466604701 -0.184718194 0.3846283869
[102,] 0.5613373000 0.051130167 -0.2170617006 -1.014313498 -0.3638828784
[103,] 0.2167733762 -0.555918283 -0.2851478923 -0.405170093 -0.4295323208
[104,] 0.4782129293 -0.494600258 -0.0680081457 -0.236930486 -0.3299819368
[105,] -0.2042113400 -0.611104506 -0.1710575170 -0.173115463 -0.2691754861
[106,] -0.1546048607 -0.359700603 -0.6403001898 -0.750351356 -1.1301517798
[107,] -0.2243220749 -0.537522876 0.4439264094 -0.141207951 0.3582609879
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[318,] -0.9509899609 -0.856376607 -0.9446638685 0.441829307 0.3631039796
[319,] -1.0354550473 -0.641763519 -0.7348847913 0.511445696 -0.3692639802
[320,] -0.2873357108 -0.598840901 -0.4106473052 0.421524527 -0.0937515662
[321,] -0.4012965416 -0.831849397 -0.5950320731 0.633274377 -0.6388571822
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[341,] -0.8209405422 -0.506863863 0.0457141961 -0.036783368 -0.9697949451
[342,] -1.1427122998 1.896802726 1.3783633868 1.039369980 0.8167308643
[343,] -1.0327736160 -0.335173392 0.3625910128 -0.019379270 -0.5802031722
[344,] -0.9938928619 -0.598840901 0.5134847350 -0.097697708 -0.3359011488
[345,] -1.2379031115 -0.562050086 0.8767337688 -0.651728138 -0.0006585045
[346,] -0.9603749705 -0.543654678 -0.1681132492 -0.982405986 -1.1791198065
[347,] -1.1628230347 -0.408755023 0.5988684998 0.264887652 0.1118065239
[348,] -0.9965742932 -0.414886825 0.8874067394 -1.025916229 0.3491131148
[349,] -0.8169183952 -0.261591762 0.7195834776 -1.002710766 0.5605904170
[350,] -0.7297718775 -0.421018628 -0.2505527463 -1.611854170 0.5858815956
[351,] -0.8048519543 0.412906515 0.4214763678 0.264887652 1.0405847007
[352,] -0.9992557245 -0.286118972 0.5591208852 0.230079457 0.5110842801
[353,] -0.8316662674 -0.457809443 0.4325173719 -0.524098091 0.6574502500
[354,] -0.9898707149 1.081272991 1.5115915025 0.473736819 0.7683009478
[355,] -1.0555657821 1.547289982 1.2749459820 0.708692132 0.6407688343
[356,] -0.3409643370 0.216688835 0.1281536932 0.934945396 0.2210428911
[357,] -0.2725878385 0.161502612 -0.0967147563 0.998760420 0.5708145104
[358,] -0.5742488613 0.486488146 0.2661662440 0.865329007 0.3776329546
[359,] -0.3302386118 0.609124196 0.1999202196 1.103185003 0.5013982968
[360,] -0.8571398649 0.670442221 0.7766286652 0.772507155 0.6025630113
[361,] -0.6251960563 -0.046978674 0.4880904257 0.586863451 0.3943143702
[362,] -0.0500290395 -0.157351119 -0.9284703959 0.775407838 0.7785250413
[363,] -1.2580138463 1.516630970 1.6698458941 1.480273777 0.8635464503
[364,] -0.9764635584 1.259095264 1.5667965228 1.416458754 1.0744856422
[365,] -1.0260700377 1.449181142 2.3702135853 1.419359437 0.9012141631
[366,] -0.6667582416 0.590728789 -0.1460312411 0.531750476 -0.5016390854
[367,] -0.3007428673 0.461960936 -0.3318881429 0.894335836 0.8522461364
[368,] -0.9550121079 0.222820637 -0.9748426129 0.531750476 0.3275887075
[369,] -0.7351347401 -0.163482922 -1.2000790959 0.505644330 0.0692958194
[370,] -0.7458604654 0.946373335 0.3445573728 1.271424610 0.0203277927
[371,] -0.6292182032 -0.249328157 -0.0385654682 -0.059988831 -0.1884589585
[372,] -0.6171517623 0.793078272 0.7935582048 0.436027941 0.5982581298
[373,] -0.1680120173 -0.059242279 0.4413501751 0.151761019 0.4050765739
[374,] -0.4388365800 0.786946469 -0.6502370934 0.169165117 0.5180797125
[375,] -0.3892301007 0.400642910 0.4523911792 0.102449410 0.5627428577
[376,] -0.6922318391 0.958636940 0.4255247360 0.505644330 0.7290189044
[377,] -0.5688859987 -0.016319661 -0.4349375142 -0.558906286 -0.2019117131
[378,] -0.4468808739 0.363852095 -0.8022349161 0.351908138 0.9006760530
[379,] 0.0263917529 0.412906515 0.0567552002 0.160463068 0.5761956123
[380,] 0.0773389479 -0.175746527 -0.5100163418 -1.498727538 -0.3778737432
[381,] -0.8276441205 -0.310646182 -0.4452424513 -0.497991945 -0.7680036262
[382,] -0.8155776795 1.829352899 1.3492887428 0.650678474 0.5681239595
[383,] -0.7834005038 0.891187112 0.7162711764 0.418623844 0.6536834787
[384,] -0.8826134624 1.412390327 0.5616971194 0.697089400 1.0341273785
[385,] -0.8759098841 1.148722818 0.1925595502 0.273589700 -0.3030764276
[386,] 0.1939812101 0.002075746 -1.0569140765 -0.889584134 -0.0996707783
[387,] -0.5957003118 0.057261969 -0.2656421185 0.354808821 0.2829255622
[388,] 0.1980033570 0.553937974 0.2588055746 -0.123803854 0.0612241666
[389,] 0.4889386546 0.652046814 -0.5427713205 0.444729990 0.5858815956
[390,] 0.5975366228 -0.279987170 -0.8548637021 -0.625621992 -0.6619959201
[391,] 0.1658261813 0.198293427 -0.8751055429 -1.008512132 -0.4903387715
[392,] 0.2395655424 0.394511108 -0.8651686392 -0.176016146 -0.3767975228
[393,] 0.4031328525 -0.427150430 -0.2391437087 -1.539337098 -0.7115020570
[394,] 0.5801073192 -0.022451464 -0.9950844537 -1.089731252 -0.3283676063
[395,] 0.1041532610 0.235084242 -0.7926660459 0.125654873 0.0596098361
[396,] 0.2543134146 0.118579994 -0.9634335754 -1.156446959 -0.8304244075
[397,] 0.2181140919 0.173766217 -0.5280499818 0.015428924 0.1440931349
[398,] -0.6841875452 2.013306974 1.6816229651 0.415723161 0.7822918126
[399,] 0.0237103216 -0.218669144 -0.4654842921 -0.329752338 -0.0878323542
[400,] -0.6493289381 -0.378096010 0.1130643209 -0.451581019 0.8005875588
[401,] -0.3838672380 -1.021935275 0.5536003831 0.273589700 0.0321662167
[402,] 0.2690612868 -1.126175918 -0.4084391044 -0.918590963 0.2301907643
[403,] -0.5393902542 -0.580445493 0.0803093422 -0.683635649 0.6967322934
[404,] -0.6345810659 -0.145087514 -0.3020774319 0.230079457 1.0760999728
[405,] -0.4602880305 -0.402623220 -0.3017093984 -0.660430186 0.5342230180
[406,] -0.3624157875 -0.261591762 0.2602777085 -0.178916829 0.5670477392
[407,] 0.2690612868 -0.316777985 0.4395100078 -0.834471159 0.4922504236
[408,] -0.4817394810 -0.120560304 0.2547572064 -0.170214780 0.8995998326
[409,] -0.2350478001 -0.739872359 -0.4765252961 -0.709741795 0.9189717992
[410,] -0.6359217815 -0.378096010 -0.2822036246 0.044435753 0.8172689745
[411,] -0.7793783568 -0.850244804 0.8023910081 -0.631423358 0.9593300630
[412,] -0.9697599801 -0.304514380 0.6905088335 0.102449410 1.1471305170
[413,] -0.4643101775 -0.347436997 0.6003406337 0.032833021 0.9426486473
[414,] 0.0263917529 0.290270465 0.2385637338 0.369312235 0.6407688343
[415,] 0.0344360469 -0.053110476 -0.0768409490 0.227178774 0.6106346640
[416,] -0.0004225602 0.044998364 -1.0639067124 1.085780906 0.3523417759
[417,] 0.1765519065 -0.353568800 0.5193732705 -0.286242095 0.8888376289
[418,] -0.7150240053 0.228952440 0.9426117597 0.282291749 0.8420220430
[419,] -0.0379625986 -0.114428502 -0.8062832842 1.088681589 0.5605904170
[420,] -0.9992557245 1.835484701 2.4924006970 0.717394181 0.8904519595
[421,] -0.9214942164 1.357204104 1.2738418816 0.479538185 1.1304491013
[422,] -0.5541381265 1.535026377 1.0276274910 0.366411552 0.8011256690
[423,] 0.2207955232 -0.316777985 -1.1043903940 -0.367461216 0.1166495155
[424,] 0.1698483282 0.014339351 -0.7753684728 0.218476726 0.1424788043
[425,] 0.1014718297 0.474224541 -0.0639597776 0.192370580 0.6735935555
[426,] 0.0210288903 0.339324885 0.8097516774 0.450531356 0.5460614420
[427,] -0.1251091163 -0.200273737 -0.4658523255 0.351908138 -0.0243353526
[428,] -0.2028706244 -0.243196355 -0.7415093937 -0.164413414 0.2850780030
[429,] 0.3535263732 -0.169614724 -0.3845169289 0.238781506 0.4190674387
[430,] 0.2033662197 -0.132823909 -0.0297326650 0.203973311 0.4831025505
[431,] 0.3508449419 0.302534070 -1.0410886373 0.369312235 -0.4182320069
[432,] 0.2744241495 0.529410763 -0.4209522423 -0.036783368 -0.0334832257
[433,] 0.2784462964 -0.365832405 -0.9465040358 0.917541299 0.6031011215
[434,] 0.7423339137 0.602992394 -1.3200580067 -0.355858484 0.0633766074
[435,] 0.2261583858 -0.696949741 -1.0388804365 0.201072628 -0.2245123408
[436,] 0.4728500667 -0.690817939 -1.1338330715 -0.570509017 0.5008601866
[437,] 0.3146456192 -0.549786481 -1.1209519001 -0.570509017 0.5390660096
[438,] -0.9925521463 2.497719374 1.4836209589 1.210510270 1.0421990312
[439,] -0.6024038901 1.516630970 0.0655880035 0.462134087 1.0481182433
[440,] 0.9018790769 -0.880903817 -0.5096483083 -0.103499074 -0.4510567281
[441,] 0.7530596390 -0.696949741 -0.3359365110 -0.570509017 -0.0205685813
[442,] 0.7785332364 -0.721476951 0.0309928574 -1.553840513 -0.0582362941
[443,] 0.5398858495 -0.071505884 -0.0466622045 -0.329752338 -0.1879208484
[444,] 0.6645724056 -0.788926779 -0.4640121582 -1.081029204 0.0983537693
[445,] -0.8504362866 1.185513634 0.7600671592 0.833421496 0.8447125939
[446,] 0.2771055808 0.290270465 -0.1092278942 -2.287713281 0.1365595923
[447,] 0.4567614788 -0.611104506 -0.6539174281 -2.067261382 0.4900979829
[448,] 0.2006847883 -0.341305195 -0.9527606048 0.372212918 -0.7599319735
[449,] -0.6453067911 -0.292250775 -0.3057577666 -0.547303554 0.1758416357
[450,] -1.0059593028 -0.212537342 0.2749990472 -0.930193694 0.1064254220
[451,] -0.6895504078 -0.568181888 -0.7054421137 -0.321050290 0.2452578494
[452,] 2.0106509251 0.768551062 -1.4683754946 -1.800398558 0.4131482267
[453,] 1.4019660169 0.811473680 -1.0101738259 -1.310183151 0.4987077458
[454,] 1.0144991920 0.762419259 -0.0301006984 -0.930193694 -0.1594010086
[455,] 0.6712759839 0.216688835 0.0081747823 -0.933094377 -0.4225368884
[456,] 1.9114379665 0.241216045 -0.9943483868 -1.942532019 -2.0201860234
[457,] 1.7425077937 0.296402268 -0.5906156715 -2.264507818 -0.7836088216
[458,] 1.4716832310 0.014339351 -0.3109102352 -2.911360100 -0.6431620636
[459,] 2.3163340948 0.094052784 -1.2538119823 -2.061460017 0.2931496557
[460,] 0.4849165076 -0.298382577 -1.0028131566 -0.956299840 -1.0042339968
[461,] -0.9724414114 2.307633495 1.4070699974 0.282291749 0.7187948109
[462,] 1.4730239467 1.283622474 -1.0745796830 -1.153546276 0.3265124871
[463,] -0.4455401583 -0.040846871 -1.0517616079 -0.962101206 -0.2089071455
[464,] 0.1671668969 0.351588490 -0.8681129070 -1.597350756 0.1882181699
[465,] -0.5420716856 0.345456688 -1.2089118991 -1.817802655 0.2210428911
[466,] -0.9496492452 -0.206405539 -1.1713724853 -1.081029204 -1.0349062773
[467,] -0.9456270983 0.112448192 -1.0326238675 -0.013577905 -0.4424469652
[468,] 0.3709556768 -0.102164897 -0.8655366727 -0.167314097 -0.3821786246
[469,] -0.7713340629 -0.635631716 -1.4525500555 -0.669132235 -0.3068431989
[470,] -0.6412846442 -0.470073048 -1.4197950768 -2.157182551 -0.6367047414
[471,] -1.1695266130 2.068493197 1.0912972811 0.407021113 1.1847982299
[472,] -0.7632897689 0.927977927 -0.7393011929 -0.065790196 0.7182567007
[473,] 1.4193953204 0.327061280 -1.0594903108 -0.335553704 -0.6103373424
[474,] 0.5050272424 0.087920982 -0.8000267152 -1.371097492 -0.1416433726
[475,] 2.5925215201 -0.831849397 -0.8430866311 -2.731517761 -0.8664777898
[476,] 2.3485112706 -0.132823909 -0.2269986043 -2.513966545 0.0133323603
[477,] 1.1110307193 -0.059242279 0.6080693365 -0.242731852 0.5218464837
[478,] 0.8415468723 -0.580445493 -1.5224764146 -0.947597791 -1.9480792588
[479,] 2.0790274236 -0.065374081 -1.0948215238 -0.996909400 0.3797853953
[480,] 0.7007717283 -0.255459960 -0.8990277184 -1.400104320 -1.1888057898
[481,] 2.8941825429 -1.868124023 -1.9953994220 -2.743120493 -0.4010124810
[482,] 2.5630257757 -0.905431027 -1.8695319757 -1.278275639 -0.3611923275
[483,] 2.2828162034 -1.107780510 -1.7738432738 -0.596615163 0.7715296089
[484,] 3.0068026581 -2.033682692 -1.9563878743 -0.695238381 -1.7215348715
[485,] 0.4125178621 1.688321440 1.2583844760 0.369312235 0.7279426840
[486,] -0.2833135638 0.179898020 -0.0455581041 -0.750351356 0.2635535956
[487,] 3.3272337001 -1.739356170 -1.0230549974 -2.830140979 -2.8790098763
[488,] -0.0191925794 0.989295953 -0.1235811995 -0.724245210 -0.3208340637
[489,] 1.5494447391 0.964768743 -0.5405631197 -0.437077605 0.4588875923
[490,] 1.8524464776 0.633651406 -1.0429288047 -0.141207951 -0.4758097966
[491,] 1.6325691098 1.167118226 -0.2641699846 -0.396468045 0.3292030380
[492,] 3.2601979173 -2.248295780 -1.4529180889 -0.880882085 -4.1403401465
[493,] 3.1810956935 -2.879871440 -1.8533385030 -1.255070176 -3.3046550315
[494,] 0.2543134146 1.069009386 0.3957140250 0.784109887 0.3458844537
[495,] 1.0225434860 0.216688835 -0.9056523208 -2.708312298 -0.8336530686
[496,] 1.8470836149 -1.181362141 -1.4713197624 -1.260871542 -2.2553401736
[497,] 1.6835163048 -2.058209902 -1.6888275425 -3.210130436 -2.4678936961
[498,] 3.4009730612 -2.554885906 -1.9983436898 -1.046221009 -3.4499447810
[499,] 2.4383392196 -3.309097617 -1.8018138174 -1.484224124 -2.4711223572
[500,] 2.7252523702 -2.935057663 -2.0182174971 -1.110036033 -3.9799833118
[501,] 0.2100697979 0.241216045 -1.3645900564 -0.927293011 -0.4758097966
[502,] 0.4996643798 -0.255459960 -0.5148007769 -0.834471159 -0.8944595194
[503,] 2.3565555645 0.474224541 -1.3016563333 -0.938895743 0.4023860230
[504,] 3.4371723840 0.265743255 -1.6991324796 -1.043320326 0.1677699830
[505,] 2.2130989893 -1.573797502 -2.1058094627 -3.381270726 -0.6463907247
[506,] 3.3393001410 -4.302449625 -2.3365664476 -4.126746226 -3.9202530815
[507,] 3.5471110678 -3.069957318 -2.1830964911 -1.028816912 -4.0192653552
[508,] 3.2829900834 -3.131275343 -2.1485013450 -1.713378071 -4.0176510247
[509,] 2.8123988879 -3.327493024 -2.0833594211 -1.208659250 -4.1285017225
[510,] 3.2159543006 -3.848696239 -1.8080703864 -1.101333984 -3.5392710715
[511,] 2.2171211362 -2.144055137 -1.8069662860 -1.104234667 -3.5026795790
[512,] 3.9399407553 -3.278438604 -1.8389851978 -1.150645593 -4.1349590447
[513,] 3.7964841800 -3.180329764 -2.0800471198 -1.246368128 -4.1543310113
[514,] -0.1184055380 1.921329936 1.6974484043 0.250384237 0.4362869646
X6 X7 X8 X9
[1,] 0.147333815 0.2466868712 0.2635143810 -0.4240896612
[2,] -0.417574672 1.2516282136 -1.0340894742 -0.4213274606
[3,] -0.349429003 0.5311042322 -1.3467651020 -0.3677762810
[4,] 0.100205035 0.5197275378 0.0243175257 -0.3856998469
[5,] -0.125885735 0.7851837414 -1.5609479070 -0.2824459650
[6,] 0.316105798 -0.9289048878 1.0686541224 -0.3342236942
[7,] 0.578498466 0.7700148155 -1.4718353531 -0.3022281709
[8,] -0.176835767 1.0013409359 -1.2185680946 -0.2442858139
[9,] 0.261334513 0.8420672137 -1.4343142778 -0.2920154830
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[504,] -2.520537269 -0.9099437304 -0.1414005570 -0.4421862574
[505,] -1.212395181 -1.1943610915 -0.4103015969 -0.4369773144
[506,] -4.571912958 -1.9186773043 4.4502410366 -0.4414750361
[507,] -5.366733468 -1.9186773043 4.0953541991 -0.4333987676
[508,] -5.422141628 -1.5773764710 4.1234950056 -0.4424328292
[509,] -1.603436681 -1.7631958136 3.7858053276 -0.4421562187
[510,] -0.006790034 -1.5622075451 3.1213696186 -0.4411448195
[511,] -0.885041221 0.2353101767 1.3656959688 -0.4395618088
[512,] -3.192440825 -1.3763882026 0.9826683248 -0.4424801521
[513,] 0.005947475 -1.6190910173 2.4303564813 -0.4409313550
[514,] 0.761918584 2.5068568337 -0.8933854417 0.0185393172
attr(,"scaled:center")
X1 X2 X3 X4 X5 X6
1.227315e+01 8.436615e+00 1.032479e+04 6.965681e+01 7.720224e+01 8.513661e+01
X7 X8 X9
5.059494e+00 6.946446e+01 2.196408e+07
attr(,"scaled:scale")
X1 X2 X3 X4 X5 X6
7.458703e+00 1.630842e+00 2.717144e+03 3.447464e+00 1.858355e+01 1.570166e+01
X7 X8 X9
2.636970e+00 6.396405e+00 4.790492e+07 > s <- cov(sc)
> s_eig <- eigen(s)
> s_eig
eigen() decomposition
$values
[1] 4.5861476 0.9947452 0.8909732 0.6612741 0.5744210 0.4566779 0.3526117
[8] 0.2667339 0.2164153
$vectors
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 0.3551323 -0.1447380 0.227288331 0.476479972 -0.08905632 -0.41925839
[2,] -0.3725700 0.1570991 0.063417197 -0.147871532 -0.36148648 -0.59662851
[3,] -0.3961092 -0.2015604 -0.175488649 -0.148879706 -0.20483856 -0.18445839
[4,] -0.3192824 -0.1290897 -0.479599915 0.007913572 0.63368823 -0.04587226
[5,] -0.3488263 0.3888999 -0.227786439 0.116152466 -0.33845783 0.12580577
[6,] -0.3102563 0.2381367 0.018848613 0.808034713 0.10041626 0.08307662
[7,] -0.3326623 -0.2414999 0.480647306 -0.022576715 0.40023302 -0.34641967
[8,] 0.3079071 -0.1917873 -0.632620046 0.154969190 -0.05955626 -0.43345751
[9,] -0.2298242 -0.7707291 -0.005834559 0.194772697 -0.35986989 0.31860139
[,7] [,8] [,9]
[1,] 0.50862254 -0.02110157 0.362474908
[2,] 0.02037587 -0.50860453 -0.254738936
[3,] -0.26905060 0.22958408 0.740559696
[4,] 0.39741885 -0.28869769 0.079766977
[5,] 0.51318596 0.49791566 -0.141557111
[6,] -0.40320376 -0.11906448 0.004668933
[7,] -0.03000319 0.49340642 -0.273749691
[8,] -0.26258624 0.28979936 -0.315942597
[9,] 0.12154478 -0.12365693 -0.232615024- Menggambar Scree Plot Berdasarkan Nilai Eigen
> plot(s_eig$values, xlab="Eigenvalue Number", ylab="Eigenvalue Size", main="Scree Plot")
> lines(s_eig$values) 
> #Membuat scree plot yang menunjukkan besarnya setiap nilai eigen untuk masing-masing komponen utama. Scree plot ini membantu menentukan jumlah komponen utama yang signifikan. Menambahkan garis untuk menghubungkan titik-titik pada plot.Interpretasi:
Berdasarkan Scree Plot tersebut, dapat diketahui secara visual bahwa dapat digunakan sekitar 3 buah PC (komponen utama). Sehingga dapat disimpulkan bahwa 3 PC pertama sudah cukup untuk menjelaskan informasi penting yang ada dalam data.
- Nilai Kumulatif Eigen
> for (eg in s_eig$values){
+ print(eg / sum(s_eig$values))
+ }
[1] 0.509572
[1] 0.1105272
[1] 0.09899702
[1] 0.0734749
[1] 0.06382455
[1] 0.05074199
[1] 0.03917908
[1] 0.0296371
[1] 0.02404615
> #Melakukan iterasi melalui semua nilai eigen dan mencetak proporsi variansi yang dijelaskan oleh masing-masing komponen utama (nilai eigen dibagi total nilai eigen). Interpretasi:
Berdasarkan proporsi kumulatif, 3 PC sudah menangkap sekitar 72% keragaman. Sehingga dapat kita susun 3 buah PC.
- Persamaan PCA
> s_eig$vectors[,1:3] #Menampilkan vektor eigen yang berhubungan dengan 3 komponen utama pertama.
[,1] [,2] [,3]
[1,] 0.3551323 -0.1447380 0.227288331
[2,] -0.3725700 0.1570991 0.063417197
[3,] -0.3961092 -0.2015604 -0.175488649
[4,] -0.3192824 -0.1290897 -0.479599915
[5,] -0.3488263 0.3888999 -0.227786439
[6,] -0.3102563 0.2381367 0.018848613
[7,] -0.3326623 -0.2414999 0.480647306
[8,] 0.3079071 -0.1917873 -0.632620046
[9,] -0.2298242 -0.7707291 -0.005834559Berdasarkan hasil di atas dapat dituliskan dalam bentuk persamaan:
PC1 = 0.355X1 - 0.373X2 + … - 0.23X9
PC2 = -0.145X1 + 0.157X2 + … - 0.771X9
PC3 = 0.23X1 + 0.063X2 + … - 0.006X9
Menampilkan Kumulatif Nilai Eigen dan Vektor Eigen Dengan Matriks Korelasi
Dekomposisi Eigen
> kor_eig <- eigen(kor) #Melakukan dekomposisi eigen pada matriks korelasi antar variabel numerik.
> kor_eig
eigen() decomposition
$values
[1] 4.5861476 0.9947452 0.8909732 0.6612741 0.5744210 0.4566779 0.3526117
[8] 0.2667339 0.2164153
$vectors
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 0.3551323 -0.1447380 0.227288331 -0.476479972 -0.08905632 0.41925839
[2,] -0.3725700 0.1570991 0.063417197 0.147871532 -0.36148648 0.59662851
[3,] -0.3961092 -0.2015604 -0.175488649 0.148879706 -0.20483856 0.18445839
[4,] -0.3192824 -0.1290897 -0.479599915 -0.007913572 0.63368823 0.04587226
[5,] -0.3488263 0.3888999 -0.227786439 -0.116152466 -0.33845783 -0.12580577
[6,] -0.3102563 0.2381367 0.018848613 -0.808034713 0.10041626 -0.08307662
[7,] -0.3326623 -0.2414999 0.480647306 0.022576715 0.40023302 0.34641967
[8,] 0.3079071 -0.1917873 -0.632620046 -0.154969190 -0.05955626 0.43345751
[9,] -0.2298242 -0.7707291 -0.005834559 -0.194772697 -0.35986989 -0.31860139
[,7] [,8] [,9]
[1,] 0.50862254 -0.02110157 -0.362474908
[2,] 0.02037587 -0.50860453 0.254738936
[3,] -0.26905060 0.22958408 -0.740559696
[4,] 0.39741885 -0.28869769 -0.079766977
[5,] 0.51318596 0.49791566 0.141557111
[6,] -0.40320376 -0.11906448 -0.004668933
[7,] -0.03000319 0.49340642 0.273749691
[8,] -0.26258624 0.28979936 0.315942597
[9,] 0.12154478 -0.12365693 0.232615024- Scree Plot
> plot(kor_eig$values, xlab="Eigenvalue Number", ylab="Eigenvalue Size", main="Scree Plot")
> lines(kor_eig$values)
Interpretasi:
Berdasarkan Scree Plot tersebut, dapat diketahui secara visual bahwa dapat digunakan sekitar 3 buah PC (komponen utama). Sehingga dapat disimpulkan bahwa 3 PC pertama sudah cukup untuk menjelaskan informasi penting yang ada dalam data.
- Nilai Kumulatif Eigen
> for (eg in kor_eig$values){
+ print(eg / sum(kor_eig$values))
+ }
[1] 0.509572
[1] 0.1105272
[1] 0.09899702
[1] 0.0734749
[1] 0.06382455
[1] 0.05074199
[1] 0.03917908
[1] 0.0296371
[1] 0.02404615Interpretasi:
Berdasarkan proporsi kumulatif, 3 PC sudah menangkap sekitar 72% keragaman. Sehingga dapat kita susun 3 buah PC.
- Persamaan PCA
> kor_eig$vectors[,1:3]
[,1] [,2] [,3]
[1,] 0.3551323 -0.1447380 0.227288331
[2,] -0.3725700 0.1570991 0.063417197
[3,] -0.3961092 -0.2015604 -0.175488649
[4,] -0.3192824 -0.1290897 -0.479599915
[5,] -0.3488263 0.3888999 -0.227786439
[6,] -0.3102563 0.2381367 0.018848613
[7,] -0.3326623 -0.2414999 0.480647306
[8,] 0.3079071 -0.1917873 -0.632620046
[9,] -0.2298242 -0.7707291 -0.005834559Berdasarkan hasil di atas dapat dituliskan dalam bentuk persamaan:
PC1 = 0.355X1 - 0.373X2 + … - 0.23X9
PC2 = -0.145X1 + 0.157X2 + … - 0.771X9
PC3 = 0.23X1 + 0.063X2 + … - 0.006X9
Fungsi PCA
prcomp
> PCA1 <- prcomp(x=Data_numeric, scale=T, center=T) #Melakukan PCA menggunakan fungsi prcomp. Argumen scale=T dan center=T memastikan bahwa data distandardisasi (dengan mean 0 dan standar deviasi 1) sebelum PCA dilakukan.
> PCA1
Standard deviations (1, .., p=9):
[1] 2.1415293 0.9973691 0.9439138 0.8131876 0.7579057 0.6757795 0.5938112
[8] 0.5164629 0.4652046
Rotation (n x k) = (9 x 9):
PC1 PC2 PC3 PC4 PC5 PC6
X1 -0.3551323 0.1447380 -0.227288331 -0.476479972 0.08905632 0.41925839
X2 0.3725700 -0.1570991 -0.063417197 0.147871532 0.36148648 0.59662851
X3 0.3961092 0.2015604 0.175488649 0.148879706 0.20483856 0.18445839
X4 0.3192824 0.1290897 0.479599915 -0.007913572 -0.63368823 0.04587226
X5 0.3488263 -0.3888999 0.227786439 -0.116152466 0.33845783 -0.12580577
X6 0.3102563 -0.2381367 -0.018848613 -0.808034713 -0.10041626 -0.08307662
X7 0.3326623 0.2414999 -0.480647306 0.022576715 -0.40023302 0.34641967
X8 -0.3079071 0.1917873 0.632620046 -0.154969190 0.05955626 0.43345751
X9 0.2298242 0.7707291 0.005834559 -0.194772697 0.35986989 -0.31860139
PC7 PC8 PC9
X1 0.50862254 0.02110157 -0.362474908
X2 0.02037587 0.50860453 0.254738936
X3 -0.26905060 -0.22958408 -0.740559696
X4 0.39741885 0.28869769 -0.079766977
X5 0.51318596 -0.49791566 0.141557111
X6 -0.40320376 0.11906448 -0.004668933
X7 -0.03000319 -0.49340642 0.273749691
X8 -0.26258624 -0.28979936 0.315942597
X9 0.12154478 0.12365693 0.232615024> print(PCA1$rotation[,1:3],digits=4) #Menampilkan vektor eigen (rotasi) dari 3 komponen utama pertama.
PC1 PC2 PC3
X1 -0.3551 0.1447 -0.227288
X2 0.3726 -0.1571 -0.063417
X3 0.3961 0.2016 0.175489
X4 0.3193 0.1291 0.479600
X5 0.3488 -0.3889 0.227786
X6 0.3103 -0.2381 -0.018849
X7 0.3327 0.2415 -0.480647
X8 -0.3079 0.1918 0.632620
X9 0.2298 0.7707 0.005835> summary(PCA1) #Menampilkan ringkasan hasil PCA, termasuk variansi yang dijelaskan oleh setiap komponen utama.
Importance of components:
PC1 PC2 PC3 PC4 PC5 PC6 PC7
Standard deviation 2.1415 0.9974 0.9439 0.81319 0.75791 0.67578 0.59381
Proportion of Variance 0.5096 0.1105 0.0990 0.07347 0.06382 0.05074 0.03918
Cumulative Proportion 0.5096 0.6201 0.7191 0.79257 0.85640 0.90714 0.94632
PC8 PC9
Standard deviation 0.51646 0.46520
Proportion of Variance 0.02964 0.02405
Cumulative Proportion 0.97595 1.00000Persamaan PCA:
Hasil di atas dapat dituliskan dalam bentuk persamaan:
PC1 = -0.355X1 + 0.373X2 + … + 0.23X9
PC2 = 0.145X1 - 0.157X2 + … + 0.771X9
PC3 = -0.23X1 - 0.063X2 + … + 0.006X9
Interpretasi Keseluruhan
Berdasarkan proportion, diketahui bahwa komponen pertama mampu menjelaskan keragaman dari 9 variabel sebesar 50.96%, komponen kedua sebesar 11.05%, dan komponen ketiga sebesar 9.9%.
Berdasarkan cumulative, terdapat 3 komponen utama yang dapat digunakan untuk analisis lanjutan karena 3 komponen tersebut dapat menjelaskan keragaman sebesar 72%.
Komponen utama yang terbentuk diambil dari matriks vektor eigen sebagai berikut:
PC1 = -0.355X1 + 0.373X2 + … + 0.23X9
PC2 = 0.145X1 - 0.157X2 + … + 0.771X9
PC3 = -0.23X1 - 0.063X2 + … + 0.006X9
3.2 Analisis Faktor
- Uji Kaiser-Meyer-Olkin (KMO) dan Measure of Sampling Adequacy (MSA)
> Data_faktor_numeric <- Data %>% select_if(is.numeric)
> library(REdaS)
> kmos <- KMOS(Data_faktor_numeric)
> kmos
Kaiser-Meyer-Olkin Statistics
Call: KMOS(x = Data_faktor_numeric)
Measures of Sampling Adequacy (MSA):
X1 X2 X3 X4 X5 X6 X7 X8
0.8624486 0.8906169 0.8355100 0.8649061 0.8436515 0.9025907 0.8018119 0.7828286
X9
0.7871056
KMO-Criterion: 0.8437785
> #Menghitung KMO (Kaiser-MeyerOlkin) untuk variabel numerik. Nilai KMO digunakan untuk menilai kelayakan data untuk analisis faktor, dengan nilai yang lebih tinggi menunjukkan data yang lebih baik untuk analisis faktor dan menyimpan hasilnya dalam variabel “kmos”.Berdasarkan output tersebut, nilai KMO sebesar 0.8437785, dimana nilai tersebut lebih besar dari 0.5, artinya indikator masih dapat digunakan meskipun dengan tingkat keakuratan yang lebih rendah. Analisis faktor dapat dilakukan.
Sedangkan nilai MSA dari seluruh variabel lebih besar dari 0.5, artinya seluruh variabel dapat digunakan dalam analisis faktor.
- Uji Barlett’s Test of Sphrecity
H0: Tidak terdapat korelasi yang signifikan antar variabel vs
H1: Terdapat korelasi yang signifikan antar variabel
> Data_barletts <- Data_faktor_numeric %>% select_if(is.numeric)
> bart_spher(x = Data_barletts) #Melakukan Uji Bartlett untuk memeriksa apakah matriks korelasi antar variabel berbeda secara signifikan dari matriks identitas (yang berarti tidak ada korelasi antar variabel). Jika hasil uji ini signifikan, maka data layak untuk analisis faktor.
Bartlett's Test of Sphericity
Call: bart_spher(x = Data_barletts)
X2 = 2160.839
df = 36
p-value < 2.22e-16Berdasarkan output tersebut, nilai p-value sebesar 2.22 x 10^-16 kurang dari alpha (0.05). Sehingga diperoleh keputusan tolak H0, artinya terdapat korelasi yang signifikan antar variabel dan layak digunakan dalam analisis faktor.
- Ekstraksi Faktor dengan PCA(Eigen Value)
> R<-cor(Data_barletts)
> R
X1 X2 X3 X4 X5 X6 X7
X1 1.0000000 -0.5439086 -0.6442219 -0.5405309 -0.5622956 -0.3735016 -0.4005907
X2 -0.5439086 1.0000000 0.6688473 0.4160388 0.6125164 0.4589014 0.5192537
X3 -0.6442219 0.6688473 1.0000000 0.5668087 0.5682333 0.4462669 0.5510882
X4 -0.5405309 0.4160388 0.5668087 1.0000000 0.4640760 0.4074537 0.4186352
X5 -0.5622956 0.6125164 0.5682333 0.4640760 1.0000000 0.5430341 0.3102468
X6 -0.3735016 0.4589014 0.4462669 0.4074537 0.5430341 1.0000000 0.4104018
X7 -0.4005907 0.5192537 0.5510882 0.4186352 0.3102468 0.4104018 1.0000000
X8 0.4623456 -0.5002928 -0.4016942 -0.2322629 -0.4391536 -0.4034359 -0.5823936
X9 -0.2415011 0.2712615 0.5128919 0.3238798 0.1866154 0.2261549 0.3934575
X8 X9
X1 0.4623456 -0.2415011
X2 -0.5002928 0.2712615
X3 -0.4016942 0.5128919
X4 -0.2322629 0.3238798
X5 -0.4391536 0.1866154
X6 -0.4034359 0.2261549
X7 -0.5823936 0.3934575
X8 1.0000000 -0.2099123
X9 -0.2099123 1.0000000
> #Menghitung matriks korelasi antar variabel numerik, yang diperlukan untuk analisis faktor berbasis PCA dan menyimpannya dalam variabel “R”. Matriks ini menunjukkan kekuatan hubungan linier antara setiap pasangan variabel.> eigen <- eigen(R) #Melakukan dekomposisi eigen pada matriks korelasi. Hasilnya adalah nilai eigen ($values) dan vektor eigen ($vectors). Nilai eigen menunjukkan proporsi variansi yang dijelaskan oleh masing-masing komponen.
> eigen$values #Menampilkan nilai eigen, yang menunjukkan berapa banyak variansi dalam data yang dapat dijelaskan oleh setiap faktor (komponen utama).
[1] 4.5861476 0.9947452 0.8909732 0.6612741 0.5744210 0.4566779 0.3526117
[8] 0.2667339 0.2164153Berdasarkan output tersebut, faktor yang memiliki nilai eigen lebih dari satu adalah komponen 1 saja (sebanyak 1 faktor), sehingga proses pemfaktoran dilakukan satu faktor saja.
- Rotasi Faktor dengan Varimax Method
> library(psych)
> pcadata<-psych::principal(Data_barletts, nfactor=1, rotate="varimax", scores=T)
> pcadata
Principal Components Analysis
Call: psych::principal(r = Data_barletts, nfactors = 1, rotate = "varimax",
scores = T)
Standardized loadings (pattern matrix) based upon correlation matrix
PC1 h2 u2 com
X1 -0.76 0.58 0.42 1
X2 0.80 0.64 0.36 1
X3 0.85 0.72 0.28 1
X4 0.68 0.47 0.53 1
X5 0.75 0.56 0.44 1
X6 0.66 0.44 0.56 1
X7 0.71 0.51 0.49 1
X8 -0.66 0.43 0.57 1
X9 0.49 0.24 0.76 1
PC1
SS loadings 4.59
Proportion Var 0.51
Mean item complexity = 1
Test of the hypothesis that 1 component is sufficient.
The root mean square of the residuals (RMSR) is 0.1
with the empirical chi square 352.6 with prob < 2e-58
Fit based upon off diagonal values = 0.95
> #principal():Melakukan analisis faktor menggunakan Principal Component Analysis (PCA).nfactors=1: Menentukan bahwa 1 faktor akan diekstrak berdasarkan jumlah faktor yang signifikan dari analisis PCA (dari nilai eigen > 1 atau berdasarkan Scree plot).rotate="varimax": Melakukan rotasi varimax, metode yang memaksimalkan variabilitas loadings pada faktor untuk membuat hasil lebih mudah diinterpretasikan.scores=T: Menghasilkan skor faktor untuk setiap observasi pada setiap faktor yang diekstrak.| Indikator | Faktor yang terbentuk |
|---|---|
| X3: Pengeluaran per kapita | Standar Hidup Layak |
| X2: Rata-rata lama sekolah | |
| X5: Persentase rumah tangga yang memiliki akses terhadap sanitasi layak | |
| X7: Tingkat pengangguran terbuka | |
| X4: Umur harapan hidup | |
| X6: Persentase rumah tangga yang memiliki akses terhadap air minum layak | |
| X9: PDRB atas dasar harga konstan menurut pengeluaran | |
| X8: Tingkat partisipasi angkatan kerja | |
| X1: Persentase penduduk miskin |
Berdasarkan hasil analisis, diketahui bahwa terbentuk satu faktor yang memengaruhi indeks pembangunan manusia tahun 2021 di berbagai kabupaten dan kota di Indonesia. Faktor tersebut adalah faktor standar hidup layak. Berdasarkan dari Proportion Variance faktor tersebut mampu menjelaskan kontribusi terhadap keragaman total sebesar 51%. Dapat disimpulkan bahwa faktor standar hidup layak memiliki pengaruh paling besar terhadap indeks pembangunan manusia di Indonesia.
4 Penutup
4.1 Kesimpulan
Penelitian memiliki total 514 kabupaten dan kota di Indonesia serta menggunakan 9 variabel yang menjadi faktor-faktor indeks pembangunan manusia tahun 2021 ini memanfaatkan analisis faktor dan PCA dalam penyelesaiannya.
Analisis terhadap faktor-faktor yang mempengaruhi indeks pembangunan manusia tahun 2021 di seluruh kabupaten dan kota di Indonesia menunjukkan bahwa tiga komponen utama mampu menjelaskan sekitar 72% dari total variasi dalam data. Dari hasil analisis faktor, berhasil diidentifikasi satu faktor utama yang memberikan kontribusi signifikan. Faktor tersebut adalah standar hidup layak. Pemberian label tersebut didasarkan pada kesamaan ciri khas variabel. Faktor ini dianggap dominan karena mampu menjelaskan 51% dari keragaman yang ada dalam data, mengindikasikan bahwa aspek standar hidup layak memiliki pengaruh besar terhadap indeks pembangunan manusia tahun 2021 di seluruh kabupaten dan kota di Indonesia. Upaya untuk memperbaiki indeks pembangunan manusia di Indonesia harus mempertimbangkan faktor standar hidup layak karena memberikan konstribusi yang signifikan.
4.2 Saran
Penelitian selanjutnya dapat fokus pada dimensi lainnya yang mempegaruhi indeks pembangunan manusia di Indonesia, seperti dimensi umur panjang dan hidup sehat serta pengetahuan. Hal tersebut dapat memberikan wawasan lebih luas terkait indeks pembangunan manusia di Indonesia. Peneliti juga dapat memperbanyak indikator/variabel indeks pembangunan manusia.
4.3 Daftar Pustaka
Setiawan, M. B., & Hakim, A. (2013). Indeks pembangunan manusia Indonesia. Jurnal Economia, 9(1), 18-26.
Simamora, B. (2005). Analisis multivariat pemasaran. Gramedia Pustaka Utama.
Subagja dan Disan. (2020). Analisis Motivasi Karyawan Terhadap Keselamatan Dan Kesehatan Kerja Menggunakan Analisis Faktor Di PT. Perkebunan Nusantara VIII. Skripsi. Universitas Komputer Indonesia.