Penerapan Analisis Multidimensional Scaling Pada Pemetaan Karakteristik IPM pada Kabupaten/Kota di Jawa Timur Tahun 2023
Tiara Kurnia Sandy
2024-11-25
- 1 Pendahuluan
- 2 Data
- 3
Source Code dan Hasil
- 3.1 Library yang dibutuhkan
- 3.2 Statistika Deskriptif Data
- 3.3 Membentuk Matriks Jarak dengan Rumus Jarak Euclidian
- 3.4 Mencari Nilai Eigen dan Vektor Eigen
- 3.5 Tititk Koordinat Objek 2 Dimensi
- 3.6 Visualisasi Objek dalam 2 Dimensi
- 3.7 Jarak Objek dari Koordinat yang Terbentuk
- 3.8 Menghitung Nilai STRESS dan Koefisien Determinasi
- 4 Hasil dan Pembahasan
- 5 Penutup
1 Pendahuluan
1.1 Latar Belakang
Indeks Pembangunan Manusia (IPM) adalah indikator ukuran apakah penduduk berhasil membangun kualitas hidupnya atau tidak. Paradigma pembangunan manusia menempatkan sebagai titik sentral (People Centered Development), sehingga setiap upaya pembangunan mempunyai ciri dari rakyat, oleh rakyat, dan untuk rakyat. Rakyat bukan hanya sebagai alat untuk mencapai hasil akhir pembangunan, tetapi sebagai tujuan akhir dari pembangunan itu sendiri. Maka dalam kerangka ini perlu diupayakan peningkatan kualitas penduduk sebagai sumber daya pembangunan, baik dari fisik (kesehatan) yang dapat dilihat dari angka harapan hidup (AHH), standar hidup layak dilihat dari presentase kemiskinan dan aspek intelektualitas (pendidikan) dilihat dari harapan lama sekolah.
1.2 Metode
Metode yang digunakan dalam analisis ini adalah Multidimensional Scaling dimana untuk mengetahui kemiripan IPM untuk wilayah Jawa Timur ditinjau dari angka harapan hidup (AMH), kemiskinan, dan harapan lama sekolah.
1.3 Tinjauan Pustaka
Multidimensional Scaling (MDS adalah suatu teknik statistik yang digunakan untuk mengukur suatu proximite (kedekatan) antar objek dalam bentuk map. Prosedur Multidimensional Scaling (MDS) menyediakan informasi yang dipahami sebagai hubungan antar objek ketika dimensi yang mendasari suatu penilaian tidak diketahui. Tujuan utama dari Multidimensional Scaling (MDS) adalah memetakan objek-objek dalam ruang multidimensional, sehingga posisi relative mencerminkan tingkat proximities antar objek (Rangkuti, 2002).
Terdapat 2 jenis penskalaan pada Multidimensional Scaling yaitu Metric Multidimensional Scaling berbasis kedekatan yang diukur dalam rasio dan Nonmetric Multidimensional Scaling beberbasis kedekatan yang diukur dalam skala ordinal. Tahapan Metric Multidimensional Scaling :
- Menghitung matriks jarak D menggunakan rumus jarak euclidian
- Setekah didapatkan matriks jarak, menghitung matriks B
- Mencari nilai eigen dan vektor eigen
- Membentuk koordinat objek
- Visualisasi dalam gambar 2 dimensi
Tahapan Nonmetric Multidimensional Scaling :
- Menghitung jarak euclidian dari koordinat yang terbentuk
- Menghitung nilai STRESS. Nilai STRESS mendekati 0 berati model memiliki tingkat kelayakan yang sempurna.
- Menentukan titik koordinat akhir.
1.4 Tujuan
Untuk mengetahui kemiripan dari kabupaten dan kota-kota yang ada di Jawa Timur dengan variabel angka harapan hidup (AHH), kemiskinan, dan harapan lama sekolah untuk nilai Indeks Pembangunan Manusia (IPM).
2 Data
Data Angka Harapan Hidup (AHH), Penduduk Miskin, dan Harapan Lama Sekolah berdasarkan Kab/Kota di Jawa Timur (2023).
Sumber : jatim.bps.go.id
3 Source Code dan Hasil
3.2 Statistika Deskriptif Data
> summary(data)
Kabupaten/Kota Se Jawa Timur Angka Harapan Hidup (Tahun) Penduduk Miskin
Length:38 Min. :67.60 Min. : 7.10
Class :character 1st Qu.:71.09 1st Qu.: 70.25
Mode :character Median :73.21 Median :114.36
Mean :72.42 Mean :110.23
3rd Qu.:73.52 3rd Qu.:152.42
Max. :74.91 Max. :251.36
Harapan lama Sekolah
Min. :11.97
1st Qu.:12.87
Median :13.48
Mean :13.52
3rd Qu.:14.01
Max. :15.77 summary(data): Memberikan ringkasan statistik (seperti rata-rata, nilai minimum, dan maksimum) untuk setiap kolom data.
3.3 Membentuk Matriks Jarak dengan Rumus Jarak Euclidian
> options(max.print = 10000)
> MJ<-as.matrix(dist(data[,-1]))
> MJ
1 2 3 4 5 6 7
1 0.0000000 7.619993 2.407571 7.697415 25.782531 94.985536 175.16228
2 7.6199934 0.000000 9.266472 14.968116 18.281450 87.470577 167.65050
3 2.4075714 9.266472 0.000000 5.821014 27.361661 96.614890 176.78748
4 7.6974152 14.968116 5.821014 0.000000 33.142087 102.383519 182.55751
5 25.7825309 18.281450 27.361661 33.142087 0.000000 69.255060 149.42621
6 94.9855363 87.470577 96.614890 102.383519 69.255060 0.000000 80.18012
7 175.1622836 167.650502 176.787478 182.557510 149.426208 80.180123 0.00000
8 17.7297716 10.559938 19.594173 25.347485 8.809024 77.408251 157.56232
9 160.2870319 152.790809 161.947965 167.721076 134.591649 65.360484 15.24608
10 43.3475074 35.881576 45.060861 50.833192 17.833645 51.696980 131.85389
11 29.4112580 22.235589 31.358477 37.048332 7.486828 66.293600 146.33958
12 7.0712729 3.815311 9.330407 14.677857 19.822058 88.623628 168.77291
13 128.9071856 121.436817 130.602849 136.380982 103.267493 34.243951 46.63194
14 77.9170245 70.440415 79.602334 85.380403 52.269473 17.287053 97.30344
15 43.0499187 35.478630 44.631938 50.366863 17.369197 52.066878 132.22613
16 36.6632214 29.162519 38.306827 44.082820 10.978930 58.323626 138.50094
17 41.1716322 33.652099 42.814550 48.580048 15.489170 53.820032 134.00005
18 39.6672674 32.180668 41.351064 47.123755 14.081207 55.330688 135.50390
19 0.8647543 7.960534 2.757861 7.534759 26.157693 95.315939 175.49291
20 13.7869902 21.223713 12.251249 6.565036 39.489510 108.690933 188.87092
21 45.1016020 37.602886 46.742752 52.520132 19.394566 49.885992 130.06154
22 77.0509195 69.552099 78.697800 84.473461 51.341230 17.957308 98.11402
23 101.0520688 93.559594 102.695909 108.475255 75.336982 6.284545 74.12534
24 73.7530528 66.231294 75.386378 81.150449 48.032607 21.243825 101.42145
25 73.5626277 66.040776 75.194062 80.958462 47.839524 21.432879 101.61119
26 120.4798763 112.998054 122.142423 127.923703 94.788941 25.653524 54.77656
27 145.5712475 138.092799 147.252312 153.030595 119.905609 50.753169 30.02250
28 50.4455687 43.040261 52.245718 57.997759 25.242462 45.024079 125.02819
29 129.9037032 122.394912 131.541404 137.311891 104.180936 34.929186 45.26700
30 55.2691406 62.712512 53.624736 47.827169 80.959107 150.167678 230.34274
31 65.6419089 73.112804 63.999717 58.213533 91.350740 160.578826 240.75654
32 38.5649763 45.977182 36.938091 31.134775 64.236159 133.420096 213.59405
33 60.3781003 67.898289 58.843802 53.096337 86.151931 155.336773 235.51107
34 62.6500798 70.161045 61.073324 55.312068 88.409080 157.622929 237.80197
35 68.5747016 76.062468 66.947240 61.169369 94.300550 163.532646 243.71209
36 67.7653422 75.253063 66.155928 60.377921 93.501468 162.722155 242.90196
37 60.2387741 52.684742 61.830325 67.577062 34.502652 34.863152 115.00781
38 69.1269072 76.614514 67.521378 61.743315 94.865042 164.082694 244.26239
8 9 10 11 12 13 14
1 17.729772 160.28703 43.347507 29.411258 7.071273 128.907186 77.917024
2 10.559938 152.79081 35.881576 22.235589 3.815311 121.436817 70.440415
3 19.594173 161.94797 45.060861 31.358477 9.330407 130.602849 79.602334
4 25.347485 167.72108 50.833192 37.048332 14.677857 136.380982 85.380403
5 8.809024 134.59165 17.833645 7.486828 19.822058 103.267493 52.269473
6 77.408251 65.36048 51.696980 66.293600 88.623628 34.243951 17.287053
7 157.562318 15.24608 131.853895 146.339585 168.772907 46.631939 97.303440
8 0.000000 142.64923 25.721353 11.855433 11.293418 111.237253 60.273274
9 142.649232 0.00000 116.948378 131.352589 153.840319 31.509702 82.376840
10 25.721353 116.94838 0.000000 14.879533 36.928153 85.563530 34.576470
11 11.855433 131.35259 14.879533 0.000000 22.631672 99.893737 49.068199
12 11.293418 153.84032 36.928153 22.631672 0.000000 122.414811 71.476529
13 111.237253 31.50970 85.563530 99.893737 122.414811 0.000000 51.001182
14 60.273274 82.37684 34.576470 49.068199 71.476529 51.001182 0.000000
15 25.756900 117.41185 3.809921 15.797183 36.880507 86.152756 35.223543
16 19.194317 123.64303 6.919176 9.581122 30.421409 92.303294 41.302621
17 23.691435 119.14276 2.876126 13.462177 34.896791 87.813480 36.818112
18 22.102581 120.62141 3.769363 11.707169 33.322269 89.258629 38.260342
19 18.030247 160.60597 43.659416 29.632045 7.144207 129.218374 78.235164
20 31.474528 174.00151 57.069883 43.024507 20.425812 142.631004 91.642787
21 27.579777 115.20540 2.560020 17.118697 38.818624 83.874270 32.877083
22 59.456662 83.25071 33.751578 48.377632 70.679464 51.961712 1.955940
23 83.441818 59.26838 57.744574 72.284538 94.665409 28.094129 23.217194
24 56.196277 86.58041 30.488883 45.166475 67.404968 55.333069 4.959143
25 56.008183 86.77296 30.302835 44.987297 67.218561 55.528379 5.145454
26 102.840316 39.83626 77.150828 91.595521 114.049693 8.800801 42.577521
27 127.911606 14.84578 102.228372 116.587248 139.097594 16.698341 67.655201
28 32.752786 110.04361 7.581992 21.314870 43.843051 78.597244 27.787654
29 112.299131 30.45803 86.588083 101.087711 123.506532 4.581637 52.042788
30 72.958703 215.48888 98.572476 84.423526 61.812690 184.128148 133.142952
31 83.327080 225.90003 108.969028 94.791542 72.177717 194.529646 143.542931
32 56.245556 198.73538 81.829167 67.710161 45.109180 167.377410 116.396042
33 77.975813 220.60222 103.662528 89.335285 66.770440 189.184969 138.233451
34 80.285367 222.91173 105.965457 91.691682 69.102253 191.508612 140.540823
35 86.247683 228.84686 111.906844 97.699124 75.090139 197.465606 146.482458
36 85.426450 228.02748 111.086946 96.852718 74.253068 196.640311 145.663320
37 42.803069 100.21046 17.270562 32.084280 53.990316 69.005129 18.271519
38 86.784488 229.38566 112.445445 98.202824 75.606678 197.996920 147.021817
15 16 17 18 19 20
1 43.049919 36.663221 41.171632 39.667267 0.8647543 13.786990
2 35.478630 29.162519 33.652099 32.180668 7.9605339 21.223713
3 44.631938 38.306827 42.814550 41.351064 2.7578615 12.251249
4 50.366863 44.082820 48.580048 47.123755 7.5347595 6.565036
5 17.369197 10.978930 15.489170 14.081207 26.1576929 39.489510
6 52.066878 58.323626 53.820032 55.330688 95.3159394 108.690933
7 132.226129 138.500939 134.000051 135.503898 175.4929144 188.870924
8 25.756900 19.194317 23.691435 22.102581 18.0302468 31.474528
9 117.411848 123.643030 119.142755 120.621411 160.6059719 174.001508
10 3.809921 6.919176 2.876126 3.769363 43.6594159 57.069883
11 15.797183 9.581122 13.462177 11.707169 29.6320452 43.024507
12 36.880507 30.421409 34.896791 33.322269 7.1442074 20.425812
13 86.152756 92.303294 87.813480 89.258629 129.2183737 142.631004
14 35.223543 41.302621 36.818112 38.260342 78.2351641 91.642787
15 0.000000 6.755568 2.696182 4.457825 43.3819559 56.684439
16 6.755568 0.000000 4.542609 3.159256 37.0036296 50.382026
17 2.696182 4.542609 0.000000 1.819341 41.5022084 54.872144
18 4.457825 3.159256 1.819341 0.000000 39.9900450 53.387143
19 43.381956 37.003630 41.502208 39.990045 0.0000000 13.444333
20 56.684439 50.382026 54.872144 53.387143 13.4443334 0.000000
21 3.367046 8.441001 4.008940 5.526518 45.4409034 58.822722
22 34.227283 40.395755 35.902138 37.391960 77.3811644 90.770141
23 58.209354 64.399955 59.910718 61.396649 101.3845767 114.777883
24 30.839707 37.094876 32.582838 34.103556 74.0801768 87.450042
25 30.647873 36.903585 32.392328 33.914783 73.8907464 87.260058
26 77.666016 83.842063 79.353760 80.822646 120.8057606 134.209720
27 102.767023 108.948426 104.455739 105.914442 145.8870498 159.295235
28 9.765598 14.458651 10.314121 11.302473 50.7176202 64.134828
29 86.989442 93.245258 88.743170 90.241131 130.2306089 143.613177
30 98.121176 91.874456 96.358866 94.887509 54.9369284 41.505908
31 108.540741 102.272238 106.764210 105.286214 65.3171310 51.900494
32 81.375859 75.137346 79.615054 78.145178 38.2193956 24.782649
33 103.363687 97.029301 101.524591 100.009888 60.0259461 46.687927
34 105.624943 99.306846 103.804012 102.301178 62.3114909 48.941530
35 111.505439 105.218874 109.714452 108.228801 68.2490798 54.845990
36 110.698327 104.410522 108.903539 107.413772 67.4308372 54.031475
37 17.220523 23.632700 19.113048 20.726394 60.5720670 73.898541
38 112.059496 105.771959 110.264289 108.773571 68.7902747 55.392167
21 22 23 24 25 26 27
1 45.101602 77.050920 101.052069 73.753053 73.562628 120.479876 145.57125
2 37.602886 69.552099 93.559594 66.231294 66.040776 112.998054 138.09280
3 46.742752 78.697800 102.695909 75.386378 75.194062 122.142423 147.25231
4 52.520132 84.473461 108.475255 81.150449 80.958462 127.923703 153.03059
5 19.394566 51.341230 75.336982 48.032607 47.839524 94.788941 119.90561
6 49.885992 17.957308 6.284545 21.243825 21.432879 25.653524 50.75317
7 130.061540 98.114024 74.125341 101.421445 101.611189 54.776555 30.02250
8 27.579777 59.456662 83.441818 56.196277 56.008183 102.840316 127.91161
9 115.205400 83.250709 59.268384 86.580410 86.772965 39.836265 14.84578
10 2.560020 33.751578 57.744574 30.488883 30.302835 77.150828 102.22837
11 17.118697 48.377632 72.284538 45.166475 44.987297 91.595521 116.58725
12 38.818624 70.679464 94.665409 67.404968 67.218561 114.049693 139.09759
13 83.874270 51.961712 28.094129 55.333069 55.528379 8.800801 16.69834
14 32.877083 1.955940 23.217194 4.959143 5.145454 42.577521 67.65520
15 3.367046 34.227283 58.209354 30.839707 30.647873 77.666016 102.76702
16 8.441001 40.395755 64.399955 37.094876 36.903585 83.842063 108.94843
17 4.008940 35.902138 59.910718 32.582838 32.392328 79.353760 104.45574
18 5.526518 37.391960 61.396649 34.103556 33.914783 80.822646 105.91444
19 45.440903 77.381164 101.384577 74.080177 73.890746 120.805761 145.88705
20 58.822722 90.770141 114.777883 87.450042 87.260058 134.209720 159.29523
21 0.000000 31.956287 55.959311 28.663895 28.472213 75.403661 100.51397
22 31.956287 0.000000 24.009719 3.548042 3.726312 43.456864 68.57376
23 55.959311 24.009719 0.000000 27.379522 27.568526 19.475703 44.61617
24 28.663895 3.548042 27.379522 0.000000 0.212132 46.828045 71.93122
25 28.472213 3.726312 27.568526 0.212132 0.000000 47.019658 72.12491
26 75.403661 43.456864 19.475703 46.828045 47.019658 0.000000 25.14856
27 100.513968 68.573762 44.616173 71.931215 72.124906 25.148557 0.00000
28 7.135678 27.166847 51.000618 24.019654 23.849667 70.294576 95.28739
29 84.806284 52.854216 28.879936 56.166731 56.357462 9.722556 16.10660
30 100.314475 132.260876 156.269437 128.926085 128.735798 175.707318 200.79166
31 110.712990 142.664853 166.671742 139.338527 139.147940 186.108408 211.19558
32 83.576203 115.515700 139.525137 112.177343 111.987543 158.960530 184.04026
33 105.466829 137.401472 161.402418 134.098857 133.910141 180.808677 205.86685
34 107.746721 139.692202 163.695909 136.383511 136.193951 183.114108 208.18536
35 113.659793 145.612346 169.618161 142.292723 142.102269 189.050314 214.13488
36 112.851457 144.800592 168.807403 141.480794 141.290851 188.234862 213.31248
37 15.280949 17.129264 41.030999 13.681181 13.487079 60.488512 85.58962
38 114.212837 146.160975 170.167951 142.841038 142.651220 189.594176 214.66987
28 29 30 31 32 33 34
1 50.445569 129.903703 55.269141 65.641909 38.56498 60.378100 62.650080
2 43.040261 122.394912 62.712512 73.112804 45.97718 67.898289 70.161045
3 52.245718 131.541404 53.624736 63.999717 36.93809 58.843802 61.073324
4 57.997759 137.311891 47.827169 58.213533 31.13477 53.096337 55.312068
5 25.242462 104.180936 80.959107 91.350740 64.23616 86.151931 88.409080
6 45.024079 34.929186 150.167678 160.578826 133.42010 155.336773 157.622929
7 125.028186 45.267005 230.342740 240.756538 213.59405 235.511073 237.801966
8 32.752786 112.299131 72.958703 83.327080 56.24556 77.975813 80.285367
9 110.043607 30.458025 215.488877 225.900035 198.73538 220.602219 222.911725
10 7.581992 86.588083 98.572476 108.969028 81.82917 103.662528 105.965457
11 21.314870 101.087711 84.423526 94.791542 67.71016 89.335285 91.691682
12 43.843051 123.506532 61.812690 72.177717 45.10918 66.770440 69.102253
13 78.597244 4.581637 184.128148 194.529646 167.37741 189.184969 191.508612
14 27.787654 52.042788 133.142952 143.542931 116.39604 138.233451 140.540823
15 9.765598 86.989442 98.121176 108.540741 81.37586 103.363687 105.624943
16 14.458651 93.245258 91.874456 102.272238 75.13735 97.029301 99.306846
17 10.314121 88.743170 96.358866 106.764210 79.61505 101.524591 103.804012
18 11.302473 90.241131 94.887509 105.286214 78.14518 100.009888 102.301178
19 50.717620 130.230609 54.936928 65.317131 38.21940 60.025946 62.311491
20 64.134828 143.613177 41.505908 51.900494 24.78265 46.687927 48.941530
21 7.135678 84.806284 100.314475 110.712990 83.57620 105.466829 107.746721
22 27.166847 52.854216 132.260876 142.664853 115.51570 137.401472 139.692202
23 51.000618 28.879936 156.269437 166.671742 139.52514 161.402418 163.695909
24 24.019654 56.166731 128.926085 139.338527 112.17734 134.098857 136.383511
25 23.849667 56.357462 128.735798 139.147940 111.98754 133.910141 136.193951
26 70.294576 9.722556 175.707318 186.108408 158.96053 180.808677 183.114108
27 95.287391 16.106601 200.791662 211.195576 184.04026 205.866853 208.185364
28 0.000000 79.778597 105.606546 115.997358 88.86542 110.602486 112.940858
29 79.778597 0.000000 185.092522 195.504773 168.34243 190.245816 192.540083
30 105.606546 185.092522 0.000000 10.457098 16.76176 6.574930 8.035832
31 115.997358 195.504773 10.457098 0.000000 27.20165 6.460418 3.879832
32 88.865421 168.342431 16.761757 27.201651 0.00000 22.237527 24.379764
33 110.602486 190.245816 6.574930 6.460418 22.23753 0.000000 2.652791
34 112.940858 192.540083 8.035832 3.879832 24.37976 2.652791 0.000000
35 118.921580 198.457227 13.466158 3.058774 30.17964 8.784777 6.186809
36 118.083934 197.644251 12.670355 2.475439 29.35826 7.827298 5.281600
37 11.901517 69.778823 115.341588 125.760344 98.59624 120.573845 122.840001
38 119.437112 199.004102 14.026454 3.767904 30.71534 9.094861 6.595605
35 36 37 38
1 68.574702 67.765342 60.23877 69.126907
2 76.062468 75.253063 52.68474 76.614514
3 66.947240 66.155928 61.83033 67.521378
4 61.169369 60.377921 67.57706 61.743315
5 94.300550 93.501468 34.50265 94.865042
6 163.532646 162.722155 34.86315 164.082694
7 243.712091 242.901964 115.00781 244.262389
8 86.247683 85.426450 42.80307 86.784488
9 228.846856 228.027478 100.21046 229.385662
10 111.906844 111.086946 17.27056 112.445445
11 97.699124 96.852718 32.08428 98.202824
12 75.090139 74.253068 53.99032 75.606678
13 197.465606 196.640311 69.00513 197.996920
14 146.482458 145.663320 18.27152 147.021817
15 111.505439 110.698327 17.22052 112.059496
16 105.218874 104.410522 23.63270 105.771959
17 109.714452 108.903539 19.11305 110.264289
18 108.228801 107.413772 20.72639 108.773571
19 68.249080 67.430837 60.57207 68.790275
20 54.845990 54.031475 73.89854 55.392167
21 113.659793 112.851457 15.28095 114.212837
22 145.612346 144.800592 17.12926 146.160975
23 169.618161 168.807403 41.03100 170.167951
24 142.292723 141.480794 13.68118 142.841038
25 142.102269 141.290851 13.48708 142.651220
26 189.050314 188.234862 60.48851 189.594176
27 214.134876 213.312476 85.58962 214.669874
28 118.921580 118.083934 11.90152 119.437112
29 198.457227 197.644251 69.77882 199.004102
30 13.466158 12.670355 115.34159 14.026454
31 3.058774 2.475439 125.76034 3.767904
32 30.179640 29.358259 98.59624 30.715340
33 8.784777 7.827298 120.57385 9.094861
34 6.186809 5.281600 122.84000 6.595605
35 0.000000 1.118794 128.72419 1.108603
36 1.118794 0.000000 127.91737 1.373499
37 128.724190 127.917365 0.00000 129.278570
38 1.108603 1.373499 129.27857 0.000000data[,-1]: Menghapus kolom pertama (Kabupaten/Kota Se Jawa Timur) agar hanya menggunakan data numerik.
dist(): Menghitung matriks jarak berbasis Euclidean distance antar baris data.
as.matrix(): Mengonversi hasil matriks jarak ke dalam bentuk matriks numerik.
MJ: Matriks jarak ini menjadi input utama untuk analisis MDS.
3.4 Mencari Nilai Eigen dan Vektor Eigen
> A <- MJ^2
> I <- diag(38)
> J <- matrix(rep(1,38),nrow = 38, ncol = 38)
> V <- I-(1/38)*A
> B <- (-1/2)*V*A*A
> eigen(B)
eigen() decomposition
$values
[1] 9.393504e+12 2.776772e+11 3.569172e+10 1.530827e+09 2.540668e+08
[6] 9.632617e+07 1.739154e+07 1.957142e+05 6.904849e+03 3.084965e+03
[11] 2.151489e-03 -2.661799e-04 -2.661799e-04 -2.661799e-04 -2.661799e-04
[16] -2.661799e-04 -2.661799e-04 -2.661799e-04 -1.211496e-01 -9.842576e-01
[21] -8.500791e+00 -1.390453e+02 -3.039037e+02 -1.785710e+03 -3.488216e+03
[26] -5.132831e+03 -5.657926e+04 -7.856538e+04 -3.504706e+06 -8.902218e+06
[31] -2.491641e+07 -6.295889e+07 -1.550681e+08 -1.282340e+09 -5.035276e+09
[36] -1.544452e+10 -3.076861e+11 -9.379068e+12
$vectors
[,1] [,2] [,3] [,4] [,5]
[1,] -0.034898918 -0.205746439 0.044058327 0.0093310103 0.073074323
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[8,] -0.018409914 -0.134773015 0.133818939 0.0619584339 -0.131349824
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[21,] 0.093373010 0.1726104074 0.071121519 -0.136802062 0.045190312
[22,] 0.017534296 0.1583444339 0.044148993 -0.072695213 0.257117014
[23,] 0.219984325 0.2104491398 0.338813923 -0.099376243 -0.140194429
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[26,] 0.289192023 0.0526398952 -0.309472892 0.052113770 -0.157663150
[27,] 0.244459314 -0.3283488785 0.046479470 -0.042138030 0.185150541
[28,] -0.284486672 -0.1594902579 -0.071298960 -0.259938266 -0.188234802
[29,] -0.217646251 -0.0351258944 -0.044927874 -0.059618788 0.095167662
[30,] 0.040718636 0.2738265534 0.033762757 0.194000187 0.103570689
[31,] -0.233073986 0.1133459984 0.128249493 -0.179476974 0.120775126
[32,] 0.171459608 -0.2543997080 -0.037210270 -0.036984462 -0.318850001
[33,] 0.056265552 0.0879991166 0.244186311 -0.064290479 -0.114820519
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[36,] 0.169189615 0.0174191818 -0.115951908 -0.178665287 0.187367075
[37,] 0.067047997 -0.2487636599 0.086998654 0.181684640 -0.234613413
[38,] 0.382269662 0.0328508912 -0.051741042 0.061423232 -0.026364439
[,11] [,12] [,13] [,14] [,15]
[1,] -0.125793730 -0.184951051 -0.1356973712 -0.1681511166 0.3671792040
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[3,] 0.070553574 0.125358586 0.0692178411 0.0272743320 -0.1433438352
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[8,] 0.039981615 0.007399313 0.0234075729 0.0532986569 -0.0679073911
[9,] -0.007576138 0.030387195 0.0019769356 -0.0053852700 -0.0013301033
[10,] -0.178219414 -0.041164655 -0.1471781734 -0.5109382269 -0.1057519226
[11,] -0.001224221 -0.015837695 0.0187825926 0.0069938348 -0.0097997705
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[,16] [,17] [,18] [,19] [,20]
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[25,] 0.136831697 -0.0900808026 0.283515118 -0.026214691 -0.008893957
[26,] -0.003085262 -0.0395168240 0.100206085 -0.150488706 0.123046043
[27,] -0.052295364 0.0186292939 -0.137952998 0.081688280 -0.055916931
[28,] -0.130434912 0.1436836158 -0.180691991 -0.217301794 -0.154963775
[29,] -0.062975506 0.0128245510 -0.037722011 0.054642752 -0.062062872
[30,] 0.010460944 0.0804712875 0.047198378 -0.025899347 0.203238436
[31,] 0.087651993 -0.1908107799 -0.206548183 -0.124573641 -0.108586028
[32,] -0.047304045 0.0009465821 0.022589588 0.162324319 -0.169982984
[33,] -0.147871831 -0.1464077560 -0.052896961 -0.130738957 -0.059060361
[34,] 0.275297921 0.1481556405 0.025608858 0.137239177 0.121752083
[35,] -0.195370922 0.0197449174 0.070321479 0.012781748 0.011214183
[36,] -0.006074506 0.0783851085 0.133626576 0.010149315 -0.114103782
[37,] -0.165454229 0.1493143719 -0.113134905 -0.263666151 0.150476999
[38,] 0.019060539 0.0171165482 -0.033330841 0.057992123 0.067801519
[,26] [,27] [,28] [,29] [,30]
[1,] 0.114062420 -1.376538e-02 -0.194516361 0.055376279 0.19399114
[2,] 0.312543427 2.583259e-01 0.040154061 0.026061611 -0.18444801
[3,] -0.192770184 -1.175415e-01 0.095262806 0.060627401 0.32068314
[4,] -0.059493175 8.276970e-02 0.188716305 0.095078569 0.14227310
[5,] 0.307543364 -4.090517e-01 0.082611052 -0.249881806 -0.05124673
[6,] 0.158391213 8.127319e-02 -0.084704570 -0.223050732 0.15751026
[7,] -0.001534882 1.058996e-02 -0.019799934 -0.034721949 0.03880776
[8,] -0.045261803 -1.785325e-02 -0.076396511 -0.225666869 0.16145004
[9,] 0.027683818 -7.206247e-02 0.054735780 0.182240977 -0.20646206
[10,] -0.322423454 3.319049e-01 -0.023075237 -0.070009948 -0.02843117
[11,] -0.122029108 -3.300773e-01 0.082747189 0.085247502 0.03297831
[12,] 0.091731672 2.108317e-02 0.239548928 0.055377926 -0.11435368
[13,] -0.001502346 -3.778201e-02 -0.125899520 0.032796488 -0.15776648
[14,] -0.187905022 -8.052402e-02 0.326212739 -0.076985152 -0.06862253
[15,] 0.153474427 -1.787037e-02 0.038425368 0.432667207 -0.15057072
[16,] 0.009993742 -1.223944e-02 -0.039874072 -0.158296454 -0.12157883
[17,] 0.098846032 1.665473e-01 0.001749890 0.035481861 -0.12316820
[18,] -0.157643967 2.641188e-01 -0.026987554 -0.075735660 -0.08068573
[19,] 0.058855388 9.406570e-02 -0.143340544 0.062593328 -0.02376933
[20,] -0.283104438 2.787593e-02 -0.160833293 0.050792892 -0.19125265
[21,] -0.093781829 -7.735658e-02 -0.093285523 -0.189533798 -0.14900284
[22,] 0.007700999 -1.222853e-02 -0.004765814 -0.104137915 -0.10801000
[23,] 0.018930393 1.861687e-01 -0.372063733 0.128461373 -0.23855279
[24,] 0.149332175 1.043961e-05 0.201624076 -0.045022534 0.19001042
[25,] 0.122293187 -1.229340e-02 0.142936884 -0.047813381 0.16851907
[26,] 0.035517773 -2.411148e-01 0.264028709 0.323071548 -0.06137451
[27,] -0.026293970 1.479068e-01 -0.032990204 -0.279228246 0.34063772
[28,] 0.269973106 6.329356e-02 -0.218242244 0.160015402 0.23792602
[29,] -0.082281779 4.001435e-02 0.014862813 -0.104252713 0.03399341
[30,] 0.220830431 4.863726e-02 -0.079366900 0.161220499 0.22665790
[31,] -0.107202783 1.870903e-01 0.140033268 0.156282885 0.00459990
[32,] -0.081072675 -1.699992e-01 0.048226143 -0.318451611 -0.25248932
[33,] -0.026644710 2.426452e-01 0.299285445 0.061836634 0.04240090
[34,] 0.128521395 -2.665026e-01 -0.374625052 -0.016946519 -0.07182576
[35,] 0.095994247 1.008059e-02 0.145633669 -0.188642646 -0.25968379
[36,] -0.170187952 -1.274366e-01 -0.124777097 0.001340413 0.06984192
[37,] -0.413871721 -2.422484e-01 -0.180676706 0.245110000 0.13718792
[38,] -0.007215612 -6.456398e-03 -0.030590665 -0.002875699 0.14389074
[,31] [,32] [,33] [,34] [,35]
[1,] -0.11552628 -0.070630408 0.009020036 0.0657886657 -0.14923994
[2,] 0.08100349 0.132528426 -0.026814545 -0.0158208353 -0.10335662
[3,] 0.13068276 0.074901636 0.190299473 0.0018822246 -0.17967140
[4,] 0.27823503 0.102495945 0.162863080 -0.0006046008 -0.20266294
[5,] -0.06448658 0.132395757 0.173135034 -0.1117885391 0.07004755
[6,] -0.01393644 -0.111501094 -0.262465706 0.0083604372 -0.16085062
[7,] -0.07078721 0.070715461 0.135216566 -0.0044286872 -0.17108085
[8,] -0.11818953 -0.147587598 -0.054806939 0.0112365027 0.01478520
[9,] 0.20021857 -0.130248191 -0.380820818 -0.0116879607 0.32205139
[10,] 0.12110540 0.025166500 0.068217271 -0.0548089839 0.20877137
[11,] 0.22344859 -0.198446911 -0.224450853 0.0293863796 0.14273524
[12,] -0.30598651 -0.254519606 -0.288729753 0.1186751014 -0.07440367
[13,] -0.33520058 0.523270206 0.042815782 0.3010043744 -0.09362302
[14,] 0.17351816 0.136903863 0.005520500 0.1782974340 0.04661233
[15,] -0.17484869 0.138580379 0.077269330 -0.1456890131 0.19048138
[16,] -0.04492965 0.052737806 0.128640633 -0.1047011113 0.16665179
[17,] -0.04380861 0.065310301 0.102000161 -0.1046795104 0.19111108
[18,] 0.03390062 0.034481287 0.086743597 -0.0825406408 0.18897579
[19,] -0.14317310 -0.079761476 -0.104688061 0.0851006056 -0.14351428
[20,] 0.02395607 -0.022601616 -0.074610333 0.0894308912 -0.17358380
[21,] -0.05696135 0.004469432 0.141470454 -0.0902527042 0.20497942
[22,] 0.02177735 -0.092628864 -0.024372703 0.0854492965 0.04619705
[23,] 0.16316397 -0.256461634 0.005547607 0.0808161110 -0.20855793
[24,] -0.11452783 -0.034980418 -0.114349133 0.0236501047 0.07819728
[25,] -0.11875651 -0.050596116 -0.107911001 0.0202899361 0.07972080
[26,] 0.12520547 -0.180917963 0.304302242 0.0213427989 -0.21895977
[27,] -0.06177381 -0.235460662 0.201219753 0.1477440880 0.17759560
[28,] 0.46525016 0.153386499 -0.021181105 0.0440838497 0.23345165
[29,] 0.04730593 0.190188449 -0.053918755 -0.5582675274 -0.15926909
[30,] -0.12338018 0.009391415 -0.032380253 -0.3230992996 0.16955395
[31,] -0.16128488 -0.198106920 0.101847710 -0.2754282311 -0.03675205
[32,] 0.11310586 0.113676590 -0.168486383 -0.1057288317 0.05856691
[33,] -0.07377794 0.086761714 0.089065573 0.4065734826 0.26053583
[34,] 0.06340943 -0.137145560 0.293678068 0.1998297008 0.15835827
[35,] 0.05017541 -0.227855038 0.159104795 -0.0858154504 -0.14729752
[36,] 0.04190866 0.125951638 -0.148844629 0.0141432774 -0.07135479
[37,] -0.30565704 -0.032574402 -0.032451066 -0.0962822449 0.17823593
[38,] 0.08840202 0.287222474 -0.331264935 0.0684475184 -0.13321484
[,36] [,37] [,38]
[1,] 0.11474376 -0.195295886 -0.0345713602
[2,] 0.15601449 -0.168699113 -0.0258758643
[3,] 0.09171772 -0.200466292 -0.0367450754
[4,] 0.03320679 -0.219939006 -0.0452665986
[5,] 0.15544036 -0.113330239 -0.0111010028
[6,] -0.25338253 -0.164252756 0.0436898292
[7,] -0.21900004 0.369901900 0.5150430302
[8,] 0.18389796 -0.136238336 -0.0167808169
[9,] 0.22178252 -0.034448099 0.3494000161
[10,] 0.04380124 -0.083583422 -0.0014210307
[11,] 0.15576492 -0.106862429 -0.0091437162
[12,] 0.16713265 -0.173364411 -0.0270176537
[13,] 0.07270450 -0.256308809 0.1406534743
[14,] -0.23540447 -0.112376649 0.0214283477
[15,] 0.04173405 -0.083679006 -0.0016241668
[16,] 0.09521497 -0.091499422 -0.0048435064
[17,] 0.05956035 -0.085569477 -0.0025229723
[18,] 0.07348912 -0.087415306 -0.0032842184
[19,] 0.11510648 -0.196587853 -0.0349938190
[20,] -0.03020621 -0.239881842 -0.0563327776
[21,] 0.02568238 -0.082010859 -0.0005266928
[22,] -0.23271500 -0.110248361 0.0205857939
[23,] -0.22420997 -0.185689263 0.0549038206
[24,] -0.21656515 -0.102755527 0.0175365288
[25,] -0.21544698 -0.102353576 0.0173709380
[26,] -0.03708933 -0.245836841 0.1075133636
[27,] 0.25476506 -0.210358784 0.2319288543
[28,] -0.01645321 -0.080149611 0.0021236825
[29,] 0.07486617 -0.256991686 0.1451220097
[30,] -0.31033686 -0.122252342 -0.2010038768
[31,] -0.01434425 0.062639224 -0.2665910535
[32,] -0.33957042 -0.250397428 -0.1240170628
[33,] -0.12216264 -0.048359329 -0.2314201914
[34,] -0.07695510 -0.004217081 -0.2462475572
[35,] 0.14072015 0.134598998 -0.2881442170
[36,] 0.11349595 0.112696344 -0.2820495767
[37,] -0.11524010 -0.082696589 0.0076541425
[38,] 0.18900005 0.148320603 -0.2922775214MJ^2: Mengkuadratkan matriks jarak (menghasilkan matriks jarak kuadrat).
diag(38): Membuat matriks identitas ukuran 38x38 (karena ada 38 kabupaten/kota).
matrix(rep(1,38),nrow=38,ncol=38): Membuat matriks ukuran 38x38 yang semua elemennya bernilai 1.
*V <- I - (1/38)A: Menghitung matriks V yang digunakan dalam perhitungan MDS.
B <- (-1/2)VA*V: Membuat matriks B berdasarkan formulasi MDS klasik.
eigen(B): Menghitung nilai eigen dari matriks B untuk menemukan koordinat optimal.
3.5 Tititk Koordinat Objek 2 Dimensi
> fit <- cmdscale(MJ,k=2)
> fit
[,1] [,2]
1 -34.029825 0.1389316
2 -26.535040 -0.8267030
3 -35.670039 -1.5820577
4 -41.447008 -1.9027807
5 -8.309074 -1.6344827
6 60.931620 -1.6192286
7 141.105704 -2.5880257
8 -16.386105 1.8802642
9 126.244507 0.7756061
10 9.301043 0.9756981
11 -5.044938 4.8406875
12 -27.579458 2.8423344
13 94.834590 3.2007823
14 43.876813 1.1659107
15 8.881454 -2.6041725
16 2.620753 -0.7548983
17 7.116996 -0.8912162
18 5.630583 0.1268935
19 -34.356349 0.6167282
20 -47.751996 -0.3582649
21 11.061332 -0.7901073
22 43.015504 -0.5822547
23 67.019792 -0.5610796
24 39.691622 -1.3748868
25 39.501008 -1.4424114
26 86.448198 0.7942039
27 111.517648 2.4960436
28 16.242814 3.8117648
29 95.858176 -1.2617971
30 -89.232783 -1.4414139
31 -99.646492 -1.1457133
32 -72.479856 -1.1760916
33 -94.349194 2.5489692
34 -96.663937 1.2909843
35 -102.596541 -0.4638933
36 -101.781331 0.1048721
37 26.100000 -2.8575155
38 -103.140189 0.2483204cmdscale(MJ, k=2): Melakukan analisis MDS untuk memetakan matriks jarak
MJke dalam ruang 2 dimensi.fit: Hasil koordinat titik dalam 2 dimensi untuk setiap kabupaten/kota.
3.6 Visualisasi Objek dalam 2 Dimensi
> x_range <- range(fit[, 1])
> y_range <- range(fit[, 2])
>
> x_margin <- diff(x_range) * 0.1 # margin 10%
> y_margin <- diff(y_range) * 0.1 # margin 10%
>
> xlim <- c(x_range[1] - x_margin, x_range[2] + x_margin)
> ylim <- c(y_range[1] - y_margin, y_range[2] + y_margin)
>
> # Plot dengan batas yang disesuaikan
> plot(fit[, 1], fit[, 2], pch = 16, xlim = xlim, ylim = ylim,
+ main = "Plot 2 Dimensi IPM Jawa Timur 2023",
+ xlab = "Dimensi 1", ylab = "Dimensi 2", col = "orange")
>
> # Menambahkan label untuk setiap titik
> text(fit[, 1], fit[, 2], labels = data$`Kabupaten/Kota Se Jawa Timur`, pos = 4, cex = 0.8)
>
> # Menambahkan garis horizontal dan vertikal di titik pusat
> abline(h = 0, col = "black")
> abline(v = 0, col = "black")
range(fit[, 1]): Menentukan rentang nilai koordinat pada dimensi X.
range(fit[, 2]): Menentukan rentang nilai koordinat pada dimensi Y.
diff(x_range) & diff(y_range): Menghitung selisih antara nilai maksimum dan minimum dari masing-masing dimensi.
x_margin & y_margin: Memberikan margin sebesar 10% dari rentang dimensi untuk membuat plot lebih luas.
xlim & ylim: Menyesuaikan batas plot pada sumbu X dan Y berdasarkan margin.
plot(): Membuat scatter plot dari koordinat 2 dimensi hasil MDS.
pch = 16: Menampilkan titik sebagai lingkaran padat.
xlim & ylim: Menggunakan rentang sumbu yang telah dihitung.
main: Memberikan judul pada plot.
xlab & ylab: Label untuk sumbu X dan Y.
col = “orange”: Warna titik.
text(): Menambahkan label nama kabupaten/kota pada titik-titik plot.
pos = 4: Label ditempatkan di sebelah kanan titik.
cex = 0.8: Ukuran teks.
abline(): Menambahkan garis horizontal dan vertikal di pusat plot (X=0, Y=0) untuk referensi.
3.7 Jarak Objek dari Koordinat yang Terbentuk
> MJt<-dist(fit)
> MJt
1 2 3 4 5 6
2 7.5567356
3 2.3774159 9.1661753
4 7.6930611 14.9507433 5.7858646
5 25.7818164 18.2438582 27.3610159 33.1390203
6 94.9777200 87.4702510 96.6016669 102.3790210 69.2406958
7 175.1567581 167.6499967 176.7786057 182.5539981 149.4178204 80.1799367
8 17.7294415 10.5037400 19.5922891 25.3448274 8.8086255 77.3968807
9 160.2755972 152.7879496 161.9317110 167.7129037 134.5751638 65.3567780
10 43.3389466 35.8813808 45.0437603 50.8296202 17.8025067 51.6957464
11 29.3637566 22.2248467 31.2913486 37.0214136 7.2513731 66.2920566
12 6.9939709 3.8147927 9.2213208 14.6569119 19.7835687 88.6234534
13 128.9007851 121.4364349 130.5922425 136.3771251 103.2569374 34.2438876
14 77.9134067 70.4400423 79.5943028 85.3789861 52.2609698 17.2807251
15 42.9988663 35.4610697 44.5632167 50.3333491 17.2178553 52.0594846
16 36.6614764 29.1558821 38.2997260 44.0827090 10.9651627 58.3172725
17 41.1597143 33.6520978 42.7926119 48.5745377 15.4439653 53.8195488
18 39.6604105 32.1797559 41.3359645 47.1213241 14.0504978 55.3285969
19 0.5787117 7.9533867 2.5613360 7.5249833 26.1443777 95.3141992
20 13.7311756 21.2221266 12.1437783 6.4914102 39.4635638 108.6909314
21 45.1007270 37.5963901 46.7380815 52.5201277 19.3888007 49.8771802
22 77.0487048 69.5509740 78.6918953 84.4728344 51.3353629 17.9461008
23 101.0520417 93.5552091 102.6949065 108.4750977 75.3365129 6.1794426
24 73.7369884 66.2289311 75.3619463 81.1403474 48.0013979 21.2414036
25 73.5478350 66.0389182 75.1711767 80.9493247 47.8104672 21.4313422
26 120.4798052 112.9948647 122.1413546 127.9236390 94.7883910 25.6304582
27 145.5665588 138.0926698 147.2441724 153.0278920 119.8978922 50.7531446
28 50.4066258 43.0285971 52.1923139 57.9721618 25.1486935 45.0176092
29 129.8955543 122.3939900 131.5286057 137.3066805 104.1679168 34.9283849
30 55.2255745 62.7007565 53.5629286 47.7880026 80.9239399 150.1645090
31 65.6292411 73.1121479 63.9779408 58.2044080 91.3387262 160.5788107
32 38.4725116 45.9461442 36.8120552 31.0413552 64.1724194 133.4122123
33 60.3674961 67.8981198 58.8243880 53.0891647 86.1417648 155.3367482
34 62.6447064 70.1608639 61.0615261 55.3092166 88.4032822 157.6224261
35 68.5693657 76.0623661 66.9358418 61.1664597 94.2947335 163.5322426
36 67.7515143 75.2520572 66.1328104 60.3677166 93.4884391 162.7220854
37 60.2044398 52.6742029 61.7832060 67.5537548 34.4308025 34.8536245
38 69.1104505 76.6126917 67.4949731 61.7306718 94.8498044 164.0824379
7 8 9 10 11 12
2
3
4
5
6
7
8 157.5551827
9 15.2370990 142.6348902
10 131.8528304 25.7030700 116.9436358
11 146.3393178 11.7211852 131.3523632 14.8574999
12 168.7725469 11.2346215 153.8378484 36.9277083 22.6229529
13 46.6318165 111.2285339 31.5034030 85.5624841 99.8929895 122.4145723
14 97.3013326 60.2671519 82.3686192 34.5762934 49.0595728 71.4759330
15 132.2242510 25.6624184 117.4117081 3.6043763 15.7914639 36.8654651
16 138.4970826 19.1886622 123.6332276 6.9008136 9.4907010 30.4136949
17 133.9994519 23.6659429 119.1391721 2.8732265 13.4449749 34.8967517
18 135.5023213 22.0863960 120.6156684 3.7673254 11.6699019 33.3208711
19 175.4913172 18.0146101 160.6009347 43.6588672 29.6141966 7.1329920
20 188.8708628 31.4456696 174.0001981 57.0686316 43.0223424 20.4248660
21 130.0567999 27.5770320 115.1938164 2.4933285 17.0621740 38.8111487
22 98.1107049 59.4526299 83.2400790 33.7504390 48.3654257 70.6779772
23 74.1136350 83.4416192 59.2397979 57.7392041 72.2668967 94.6604523
24 101.4213375 56.1721240 86.5795966 30.4813479 45.1662836 67.4031390
25 101.6111547 55.9857978 86.7718523 30.2966192 44.9868714 67.2171695
26 54.7620530 102.8400381 39.7963137 77.1473688 91.5825745 114.0460481
27 30.0216722 127.9052358 14.8270119 102.2279116 116.5861649 139.0975371
28 125.0267920 32.6860374 110.0435863 7.4987637 21.3126030 43.8329928
29 45.2669597 112.2882510 30.4545583 86.5860484 101.0874809 123.5058436
30 230.3413412 72.9223701 215.4886957 98.5634684 84.4219054 61.8019665
31 240.7565165 83.3153562 225.8991703 108.9681868 94.7907752 72.1772955
32 213.5902268 56.1769542 198.7339471 81.8092024 67.7028050 45.0798571
33 235.5109296 77.9659570 220.6008298 103.6621765 89.3336567 66.7703812
34 237.8012808 80.2799950 222.9090406 105.9654492 91.6877389 69.1018960
35 243.7115019 86.2423000 228.8444052 111.9068437 97.6957207 75.0899056
36 242.9019627 85.4136793 228.0268249 111.0857871 96.8522468 74.2523515
37 115.0060198 42.7494525 100.2103884 17.2307426 32.0822300 53.9812232
38 244.2623614 86.7694320 229.3853026 112.4435846 98.2026892 75.6052449
13 14 15 16 17 18
2
3
4
5
6
7
8
9
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14 50.9983895
15 86.1489353 35.1978504
16 92.2986404 41.3007499 6.5281074
17 87.8129875 36.8173319 2.4591731 4.4983082
18 89.2569524 38.2603401 4.2458074 3.1363407 1.8016572
19 129.2167789 78.2350892 43.3576036 37.0025330 41.5007494 39.9899323
20 142.6309972 91.6414850 56.6779657 50.3743113 54.8715803 53.3847843
21 83.8682652 32.8737249 2.8359657 8.4406520 3.9456320 5.5076239
22 51.9569918 1.9488291 34.1938816 40.3951197 35.8998380 37.3916460
23 28.0680350 23.2073258 58.1742259 64.3993300 59.9037058 61.3930632
24 55.3324826 4.8960671 30.8346819 37.0760528 32.5782171 34.0941301
25 55.5280520 5.0942140 30.6415854 36.8866618 32.3887024 33.9067597
26 8.7248603 42.5730078 77.6411536 83.8417567 79.3491039 80.8203695
27 16.6979369 67.6539125 102.7628366 108.9454098 104.4555876 105.9135656
28 78.5941511 27.7603757 9.7649305 14.3671477 10.2663812 11.2337753
29 4.5784653 52.0380238 86.9870807 93.2388008 88.7419545 90.2382790
30 184.1259022 133.1351296 98.1211271 91.8561023 96.3513500 94.8763297
31 194.5296463 143.5419197 108.5377455 102.2679924 106.7637912 105.2847671
32 167.3716846 116.3802361 81.3738421 75.1017906 79.5973615 78.1213064
33 189.1849071 138.2329264 103.3591875 97.0262147 101.5244927 100.0091118
34 191.5080501 140.5408060 105.6172424 99.3057677 103.8038732 102.3011442
35 197.4651394 146.4824210 111.4985388 105.2176969 109.7143689 108.2287369
36 196.6402934 145.6620084 110.6959390 104.4056246 108.9028822 107.4119167
37 69.0010639 18.2264378 17.2204096 23.5732054 19.0845692 20.6858335
38 197.9967933 147.0198655 112.0579549 105.7657007 110.2630734 108.7708404
19 20 21 22 23 24
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20 13.4310828
21 45.4394643 58.8149137
22 77.3811424 90.7677769 31.9548482
23 101.3829823 114.7719673 55.9589285 24.0042969
24 74.0747496 87.4495279 28.6362616 3.4170832 27.3402842
25 73.8860552 87.2597391 28.4471554 3.6182255 27.5328936 0.2022213
26 120.8046771 134.2051427 75.4035118 43.4544995 19.4756196 46.8068620
27 145.8861023 159.2952189 100.5100505 68.5712743 44.6027492 71.9302588
28 50.6999358 64.1305298 6.9300056 27.1308748 50.9649221 24.0155778
29 130.2280746 143.6130150 84.7981563 52.8470414 28.8468964 56.1666681
30 54.9150164 41.4949263 100.2962302 132.2510783 156.2550551 128.9244227
31 65.3139268 51.9004699 110.7083954 142.6631091 166.6673094 139.3383028
32 38.1656391 24.7413800 83.5420797 115.4968869 139.5010035 112.1716543
33 60.0239543 46.6878023 105.4633990 137.4003821 161.3989533 134.0982370
34 62.3112368 48.9397386 107.7453694 139.6920021 163.6942069 136.3816172
35 68.2487478 54.8446464 113.6583412 145.6120933 169.6163607 142.2910795
36 67.4269251 54.0313197 112.8462122 144.7984656 168.8024365 141.4806919
37 60.5560935 73.8942731 15.1801090 17.0678381 40.9841797 13.6722487
38 68.7848270 55.3915143 114.2062423 146.1580534 170.1619060 142.8410345
25 26 27 28 29 30
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26 47.0004375
27 72.1242536 25.1271486
28 23.8442856 70.2702047 95.2839191
29 56.3574582 9.6319694 16.1040500 79.7768577
30 128.7337910 175.6952054 200.7890420 105.6063324 185.0910469
31 139.1478161 186.1048011 211.1955412 115.9952921 195.5047031 10.4179062
32 111.9811803 158.9402668 184.0341440 88.8627639 168.3380542 16.7550282
33 133.9096999 180.8059078 205.8668496 110.5992175 190.2455410 6.4885144
34 136.1923776 183.1128093 208.1850735 112.9348873 192.5390376 7.9175786
35 142.1009178 189.0489253 214.1346476 118.9162456 198.4563214 13.3994614
36 141.2908111 188.2307912 213.3123819 118.0823432 197.6442326 12.6434588
37 13.4755158 60.4585814 85.5852513 11.9014041 69.7764251 115.3414767
38 142.6512166 189.5891730 214.6696053 119.4361733 199.0040953 14.0096802
31 32 33 34 35 36
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32 27.1666532
33 6.4584860 22.1843198
34 3.8513800 24.3095919 2.6344944
35 3.0278155 30.1251049 8.7804365 6.1867100
36 2.4741667 29.3294614 7.8236989 5.2530543 0.9940128
37 125.7581431 98.5941944 120.5704711 122.8340115 128.7187985 127.9156384
38 3.7615487 30.6934029 9.0870552 6.5596482 0.8959920 1.3664088
37
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38 129.2775027MJt <- dist(fit): Menghitung jarak baru (rekonstruksi) berdasarkan hasil koordinat MDS
3.8 Menghitung Nilai STRESS dan Koefisien Determinasi
> a<-sum((MJ-MJt)^2)
> b<-sum((MJ-(sum(MJ)/38))^2)
> STRESS<-sqrt(a/b)
> cat("STRESS :",STRESS)
STRESS : 0.02723654
> RSQ<-1-(a/b)
> cat("R Square :",RSQ)
R Square : 0.9992582a: Total kesalahan kuadrat antara jarak asli (MJ) dan jarak rekonstruksi (MJt).
b: Variabilitas total dalam matriks jarak asli.
STRESS: Mengukur tingkat ketidaksesuaian (badness-of-fit). Semakin kecil nilai STRESS, semakin baik hasil MDS.
RSQ (R-Square): Mengukur proporsi variansi dalam jarak asli yang dapat dijelaskan oleh hasil MDS. Nilai mendekati 1 menunjukkan hasil MDS sangat baik.
4 Hasil dan Pembahasan
Statistika Deskriptif
Angka Harapan Hidup (AHH)
Statistika deskriptif karakteristik kesehatan pada tiap kabupaten/kota di Jawa Timur menunjukkan Kabupaten Bondowoso memiliki angka harapan hidup terendah, yaitu 67,60 tahun, Kabupaten Tulungagung memiliki angka harapan hidup tertinggi, yaitu 74,91 tahun. Rata-rata angka harapan hidup di Jawa Timur adalah 72,42 tahun, dengan nilai tengah (median) 73,21 tahun. Kuartil pertama (1st Qu.) adalah 71,09 tahun, yang menunjukkan bahwa 25% kabupaten/kota memiliki angka harapan hidup di bawah nilai ini. Kuartil ketiga (3rd Qu.) adalah 73,52 tahun, yang menunjukkan bahwa 75% kabupaten/kota memiliki angka harapan hidup di bawah nilai ini.
Standar Hidup Layak (Penduduk Miskin)
Statistika deskriptif karakteristik standar hidup layak (dilihat dari persentase penduduk miskin) menunjukkan Kota Batu memiliki persentase penduduk miskin terendah, yaitu 7,10%, Kabupaten Malang memiliki persentase penduduk miskin tertinggi, yaitu 251,36%. Rata-rata persentase penduduk miskin di Jawa Timur adalah 110,23%, dengan nilai tengah (median) 114,36%. Kuartil pertama (1st Qu.) adalah 70,25%, yang menunjukkan bahwa 25% kabupaten/kota memiliki persentase penduduk miskin di bawah nilai ini. Kuartil ketiga (3rd Qu.) adalah 152,42%, yang menunjukkan bahwa 75% kabupaten/kota memiliki persentase penduduk miskin di bawah nilai ini.
Pendidikan (Harapan Lama Sekolah)
Statistika deskriptif karakteristik pendidikan pada tiap kabupaten/kota di Jawa Timur menunjukkan Kabupaten Bangkalan memiliki harapan lama sekolah terendah, yaitu 11,97 tahun, Kota Malang memiliki harapan lama sekolah tertinggi, yaitu 15,77 tahun. Rata-rata harapan lama sekolah di Jawa Timur adalah 13,52 tahun, dengan nilai tengah (median) 13,48 tahun. Kuartil pertama (1st Qu.) adalah 12,87 tahun, yang menunjukkan bahwa 25% kabupaten/kota memiliki harapan lama sekolah di bawah nilai ini. Kuartil ketiga (3rd Qu.) adalah 14,01 tahun, yang menunjukkan bahwa 75% kabupaten/kota memiliki harapan lama sekolah di bawah nilai ini.
Matriks Jarak Euclidian
Matriks jarak menunjukkan bahwa Kabupaten Lamongan dan Kabupaten Gresik memiliki jarak terdekat sebesar 0.212132 diantara kabupaten/kota lainnya, yang berati kedua kabupaten memiliki karakteristik IPM yang mirip. Kabupaten Malang dan Kota Batu memiliki jarak terjauh sebesar 244.2624 diantara kabupaten/kota lainnya, yang berati keduanya memiliki karakteristik IPM yang sangat berbeda.
Nilai Eigen dan Vektor Eigen
Dengan jumlah 2 dimensi didapatkan nilai eigen value positif pertama dari matriks B yaitu :
\[ λ_1=9.393504e+12 \] \[λ_2=2.776772e+11 \]
Visualisasi
Dari koordinat 38 objek didapatkan plot 2 dimensi yang dapat menentukan kemiripan kota/kabupaten di Jawa Timur tahun 2023 sesuai dengan karakteristik IPM. Berikut beberapa kelompok yang berdekatan :
Kelompok 1 : Kabupaten Banyuwangi, Kabupaten Bangkalan, Kabupaten Nganjuk, Kabupaten Probolinggo, Kabupaten Sampang
Kelompok 2 : Kota Probolinggo, Kota Pasuruan, Kota Batu, Kota Madiun, Kabupaten Situbondo, Kabupaten Lumajang, Kabupaten Madiun, Kabupaten Pacitan, Kabupaten Bondowoso
Kelompok 3 : Kota Mojokerto, Kota Blitar, Kota Kediri, Kota Malang, Kabupaten Magetan, Kabupaten Trenggalek, Kabupaten Tulungaggung, Kabupaten Ponorogo, Kabupaten Blitar
Kelompok 5 : Kabupaten Sidoarjo, Kota Surabaya, Kabupaten Jombang, Kabupaten Ngawi, Kabupaten Mojokerto, Kabupaten Kediri, Kabupaten Lamongan, Kabupaten Gresik, Kabupaten Tuban, Kabupaten Bojonegoro
Kelompok Terpisah : Kabupaten Jember, Kabupaten Malang, Kabupaten Sumenep, Kabupaten Pasuruan, Kabupaten Pamekasan.
Nilai STRESS dan Koefisien Determinasi
Nilai STRESS yang diperoleh sebesar 0.02723654 atau 2.7% yang menunjukkan bahwa model penskalaan yang diperoleh termasuk kriteria hampir sempurna karena sangat kecil mendekati 0.
Nilai Rsquare yang diperoleh dalam analisis Multidimensional Scaling menunjukkan bahwa model penskalaan sangat mewakili data karena bernilai 0.9992582 atau 99.92%.
5 Penutup
5.1 Kesimpulan
Berdasarkan plot 2 dimensi dapat disimpulkan bahwa terdapat 5 kelompok kabupaten/kota yang memiliki karakteristik IPM dengan tingkatan Angka Harapan Hidup (AHH), Kemiskinan, Harapan Lama Sekolah yang berbeda. Peta geometri 2 dimensi yang terbentuk memiliki standar kelayakan yang tinggi karena nilai STRESS 2.7% yang termasuk kategori sempurna dan nilai determinasi sebesar 99.92% yang berati sangat akurat.
5.2 Saran
Peningkatan Angka Harapan Hidup, penurunan presentase kemiskinan dan harapan lama sekolah dapat disesuaikan dengan kelompok yang telah didapatkan.
5.3 Daftar Pustaka
Badan Pusat Statistik Provinsi Jawa Timur. (n.d.). Angka Harapan Hidup di Provinsi Jawa Timur. Badan Pusat Statistik Provinsi Jawa Timur.
https://jatim.bps.go.id/id/statistics-table/2/MjkjMg==/angka-harapan-hidup.html
Badan Pusat Statistik Provinsi Jawa Timur. (n.d.). Jumlah Penduduk Miskin Menurut Kabupaten/Kota di Jawa Timur. Badan Pusat Statistik Provinsi Jawa Timur.
Badan Pusat Statistik Provinsi Jawa Timur. (2024, Mei 20). Indeks Pembangunan Manusia Provinsi Jawa Timur 2023. Badan Pusat Statistik Provinsi Jawa Timur. https://jatim.bps.go.id/id/publication/2024/05/20/6db5b563181355d2c47485e1/indeks-pembangunan-manusia-provinsi-jawa-timur-2023.html
Rangkuti, Freddy. 2002. Creating Effective Marketing Plan. Jakarta : PT Gramedia Pustaka Utama.