ANALISIS MULTIDIMENSIONAL SCALING UNTUK MENGIDENTIFIKASI POLA KEMIRIPAN ANTARPROVINSI BERDASARKAN KETERSEDIAAN FASILITAS PENDIDIKAN DESA
Nayla Shabrina
2025-11-26
Library:
> # install.packages("knitr")
> # install.packages("rmarkdown")
> # install.packages("prettydoc")
> # install.packages("equatiomatic")1 PENDAHULUAN
1.1 Latar Belakang
Pendidikan merupakan salah satu pilar penting dalam pembangunan nasional. Ketersediaan fasilitas pendidikan tidak hanya menentukan kualitas sumber daya manusia, tetapi juga mencerminkan pemerataan pembangunan antarwilayah. Dalam konteks Indonesia, pemerintah telah berupaya menyediakan layanan pendidikan hingga ke tingkat desa agar seluruh masyarakat memiliki akses yang setara terhadap pendidikan dasar dan menengah.
Meskipun demikian, kondisi ketersediaan fasilitas pendidikan antarprovinsi masih menunjukkan variasi yang cukup tinggi. Beberapa provinsi memiliki jumlah desa dengan fasilitas pendidikan yang lengkap, sementara provinsi lainnya masih memiliki keterbatasan tertentu. Variasi ini penting untuk dikaji guna memahami pola penyebaran fasilitas serta kesenjangan akses pendidikan di tingkat nasional.
Untuk melihat pola kemiripan antarprovinsi berdasarkan ketersediaan fasilitas pendidikan desa, diperlukan suatu pendekatan analisis multivariat yang mampu menyajikan hubungan antarobjek secara visual dan informatif. Salah satu metode yang tepat digunakan adalah Multidimensional Scaling (MDS). MDS berfungsi memetakan objek ke dalam ruang berdimensi rendah (biasanya dua dimensi) berdasarkan jarak atau kemiripan di antara objek tersebut. Dengan menggunakan MDS, hubungan dan struktur persepsi antarprovinsi dapat digambarkan secara jelas melalui grafik dua dimensi sehingga pola kemiripan dan perbedaannya lebih mudah diinterpretasikan. Melalui analisis MDS, diharapkan dapat diketahui bagaimana pola pemerataan fasilitas pendidikan desa antarprovinsi di Indonesia serta kelompok provinsi mana saja yang memiliki karakteristik serupa maupun berbeda. Informasi ini penting sebagai dasar perumusan kebijakan pemerataan layanan pendidikan di tingkat daerah maupun nasional.
1.2 Rumusan Masalah
Berdasarkan latar belakang tersebut, rumusan masalah dalam laporan praktikum ini adalah sebagai berikut:
Bagaimana pola penyebaran desa yang memiliki fasilitas pendidikan menurut provinsi pada tahun 2024?
Bagaimana hubungan dan kemiripan antarprovinsi di Indonesia berdasarkan jumlah desa yang memiliki fasilitas pendidikan pada berbagai jenjang ketika dianalisis menggunakan metode Multidimensional Scaling (MDS)?
Bagaimana kualitas pemetaan MDS yang dihasilkan berdasarkan keragaman kumulatif dan nilai “Stress” sebagai indikator ketepatan model?
1.3 Tujuan
Sejalan dengan rumusan masalah di atas, tujuan penelitian ini adalah:
Mengetahui pola penyebaran desa yang memiliki fasilitas pendidikan menurut provinsi pada tahun 2024.
Menganalisis hubungan dan kemiripan antarprovinsi di Indonesia berdasarkan jumlah desa yang memiliki fasilitas pendidikan pada berbagai jenjang menggunakan metode Multidimensional Scaling (MDS).
Mengevaluasi kualitas pemetaan MDS yang dihasilkan melalui nilai keragaman kumulatif dan nilai “Stress” sebagai indikator ketepatan model.
2 Tinjauan Pustaka
Multidimensional Scaling (MDS) merupakan teknik analisis multivariat yang digunakan untuk memvisualisasikan tingkat kemiripan atau ketidaksamaan antarobjek dengan merepresentasikannya dalam ruang berdimensi rendah. Tujuannya adalah menghasilkan konfigurasi titik sehingga jarak antar titik dalam ruang representasi mencerminkan struktur ketidaksamaan yang diberikan. Menurut Härdle & Simar (2015), MDS mencoba “mendekatkan jarak Euclidean antartitik terhadap nilai dissimilarity dalam data” sehingga pola hubungan antar objek lebih mudah diamati secara visual. Dalam konteks analisis data kompleks, MDS membantu merangkum struktur hubungan yang tersembunyi, terutama ketika informasi yang tersedia bukan berupa variabel numerik, melainkan matriks jarak atau persepsi.
3 SOURCE CODE
3.1 Library
> # Library
> library(knitr)
> library(rmarkdown)
> library(prettydoc)
> library(equatiomatic)
> library(readxl)
> library(MASS)
> library(ggplot2)
> library(ggrepel)3.2 Impor Data
> data = read_excel("C:/Users/nayla/Downloads/Jumlah Desa yang Memiliki Fasilitas Sekolah Menurut Provinsi dan Tingkat Pendidikan, 2024.xlsx")
> head(data)
# A tibble: 6 × 6
Provinsi SD SMP SMU SMK `Perguruan Tinggi`
<chr> <dbl> <dbl> <dbl> <dbl> <dbl>
1 ACEH 3382 1421 735 205 119
2 SUMATERA UTARA 5003 2319 1147 712 202
3 SUMATERA BARAT 1256 808 415 180 102
4 RIAU 1811 1210 631 259 80
5 JAMBI 1484 810 393 169 42
6 SUMATERA SELATAN 2996 1410 720 252 994 HASIL DAN PEMBAHASAN
4.1 Matriks Jarak
Langkah pertama dalam MDS adalah membentuk matriks jarak Euclidean. Dalam kasus ini, matriks jarak menunjukkan seberapa besar perbedaan antarprovinsi dalam hal jumlah desa yang memiliki fasilitas pendidikan. Jarak ini merepresentasikan kemiripan antarobjek, semakin kecil jarak antar dua titik maka kedua objek tersebut semakin mirip.
> Data = data[,-1]
> D = as.matrix(dist(Data))
> D
1 2 3 4 5 6 7
1 0.0000 1966.654 2235.8352 1589.9104 2024.8244 389.8089 2449.9394
2 1966.6538 0.000 4141.4947 3450.3759 3943.2736 2293.2222 4385.9334
3 2235.8352 4141.495 0.0000 723.1943 237.0506 1867.6782 411.3344
4 1589.9104 3450.376 723.1943 0.0000 577.1629 1205.2203 1063.9521
5 2024.8244 3943.274 237.0506 577.1629 0.0000 1662.2909 487.3346
6 389.8089 2293.222 1867.6782 1205.2203 1662.2909 0.0000 2104.6867
7 2449.9394 4385.933 411.3344 1063.9521 487.3346 2104.6867 0.0000
8 869.7183 2677.474 1471.6192 777.3500 1285.2058 485.2350 1753.9119
9 3294.4107 5224.075 1109.4877 1829.9620 1286.9491 2942.8892 849.1142
10 3257.2556 5184.613 1063.1834 1783.8018 1245.8523 2903.7674 817.8038
11 3368.6555 5276.339 1153.1466 1865.4981 1354.8852 3008.6168 964.4911
12 4547.5411 2807.628 6416.6815 5711.4372 6264.2751 4770.6780 6738.2813
13 5654.6275 3724.856 7850.9903 7164.7465 7648.7485 6000.0426 8078.5025
14 3202.9799 5121.451 996.5094 1716.3219 1186.4696 2845.9661 774.5560
15 6595.2029 4649.018 8692.1492 7977.7884 8514.0352 6896.4795 8978.5086
16 1875.6404 3649.728 706.6385 400.7593 668.4213 1488.4885 1091.4266
17 2953.5364 4880.099 787.7512 1505.9894 952.6568 2602.7666 511.7714
18 2286.4518 4152.962 254.5958 715.7045 415.7571 1907.2050 621.1159
19 326.3219 1888.965 2278.3097 1599.3564 2075.2624 456.0669 2519.8371
20 1364.7029 3272.262 893.7869 343.6466 688.6697 996.3318 1143.2913
21 1997.6506 3925.959 336.8724 624.3485 144.2359 1643.4692 481.7043
22 1690.9468 3633.852 634.9693 556.1672 405.9372 1352.7572 760.7943
23 2582.8900 4504.384 404.8531 1118.9254 567.6839 2225.8140 210.7131
24 3393.1055 5324.466 1211.4710 1932.0461 1388.3346 3042.9009 946.4280
25 2010.1142 3937.587 362.7478 676.0399 191.1361 1661.2122 454.0418
26 1574.9727 3500.139 696.5572 405.8891 473.0507 1220.1188 897.8953
27 573.6942 2157.786 2024.9731 1315.8693 1843.3695 405.8694 2312.9440
28 1625.5433 3555.317 631.1022 390.8683 404.7159 1269.5480 839.6452
29 2994.0817 4924.292 819.9305 1537.8602 988.1361 2642.2475 551.5941
30 2989.5786 4911.663 790.0019 1509.3187 972.1420 2632.7402 562.9503
31 2459.7421 4385.176 292.8447 997.4267 444.0574 2102.2027 180.8480
32 2471.5532 4392.678 293.9966 1001.4789 451.3214 2112.4214 195.4431
33 3228.7841 5165.539 1080.1287 1798.4632 1241.0846 2884.0645 781.3783
34 3254.2145 5188.631 1091.2749 1811.2148 1258.7541 2907.0495 806.6988
35 3043.5765 4979.026 896.8238 1613.5077 1054.1594 2697.5083 595.8800
36 3210.3179 5148.483 1082.6883 1797.6880 1234.5408 2868.9109 765.7297
37 3297.9532 5232.319 1138.6668 1858.5403 1304.8199 2951.6819 849.5669
38 3197.1547 5134.343 1051.7352 1768.9689 1210.3281 2852.5902 749.1135
8 9 10 11 12 13 14
1 869.7183 3294.41072 3257.25559 3368.6555 4547.541 5654.627 3202.9799
2 2677.4744 5224.07475 5184.61291 5276.3392 2807.628 3724.856 5121.4513
3 1471.6192 1109.48772 1063.18343 1153.1466 6416.681 7850.990 996.5094
4 777.3500 1829.96202 1783.80184 1865.4981 5711.437 7164.747 1716.3219
5 1285.2058 1286.94911 1245.85232 1354.8852 6264.275 7648.749 1186.4696
6 485.2350 2942.88923 2903.76738 3008.6168 4770.678 6000.043 2845.9661
7 1753.9119 849.11424 817.80377 964.4911 6738.281 8078.503 774.5560
8 0.0000 2568.84254 2526.37527 2613.5267 5006.741 6390.293 2459.8768
9 2568.8425 0.00000 63.08724 270.6695 7509.851 8921.549 159.1666
10 2526.3753 63.08724 0.00000 242.0372 7462.205 8885.050 108.7934
11 2613.5267 270.66954 242.03719 0.0000 7500.933 8981.863 215.8425
12 5006.7407 7509.85060 7462.20517 7500.9329 0.000 2829.865 7380.6956
13 6390.2934 8921.54930 8885.05037 8981.8628 2829.865 0.000 8822.5775
14 2459.8768 159.16658 108.79338 215.8425 7380.696 8822.578 0.0000
15 7229.3222 9795.42909 9751.28325 9819.4599 2621.198 1810.597 9680.2672
16 1018.5534 1746.67312 1695.34097 1729.3406 5779.612 7356.922 1611.9525
17 2231.2810 349.70702 316.82961 469.9372 7181.108 8576.823 274.2754
18 1481.0287 1182.88715 1129.77520 1178.7001 6352.644 7869.195 1053.6555
19 839.7952 3356.46362 3318.31644 3422.7249 4378.678 5582.721 3258.0298
20 636.4621 1967.41150 1928.53208 2038.0182 5634.831 6972.048 1868.9296
21 1286.4875 1308.14296 1272.36709 1399.6214 6279.016 7620.985 1218.5885
22 1047.0439 1608.37091 1574.51421 1703.7726 6048.108 7323.651 1525.1488
23 1847.2534 721.98961 681.54750 801.5816 6803.263 8205.284 621.9317
24 2670.7031 102.65963 153.68474 283.6917 7611.704 9021.220 245.8190
25 1310.4408 1292.84763 1258.15063 1381.2541 6299.336 7629.638 1202.8421
26 879.9472 1733.72633 1697.40243 1816.4204 5880.820 7194.400 1641.2322
27 581.7517 3127.11416 3084.29652 3173.1683 4496.859 5864.136 3019.3044
28 927.0690 1674.63220 1636.74525 1753.5116 5931.774 7255.965 1581.5293
29 2270.7737 305.31459 276.33856 463.5440 7225.093 8620.280 250.4536
30 2250.4897 322.84362 279.77491 424.6787 7188.558 8612.249 223.2263
31 1727.4481 845.46614 805.16893 929.9306 6693.885 8087.154 750.0553
32 1732.8765 836.78731 795.31189 912.8324 6692.941 8096.075 736.9199
33 2523.5921 136.07718 162.58844 375.7273 7482.923 8858.015 235.0468
34 2541.7238 88.94942 118.19475 330.9441 7494.634 8883.557 201.2411
35 2336.3240 266.00376 249.21878 450.0600 7298.755 8672.996 254.3344
36 2514.1836 196.55788 223.74763 427.3231 7478.737 8837.259 285.4488
37 2587.1295 98.17841 145.49570 335.4504 7539.624 8925.535 231.0563
38 2492.8550 155.60848 176.56160 398.6201 7454.608 8825.835 239.3115
15 16 17 18 19 20 21
1 6595.203 1875.6404 2953.5364 2286.4518 326.3219 1364.7029 1997.6506
2 4649.018 3649.7279 4880.0992 4152.9622 1888.9651 3272.2618 3925.9594
3 8692.149 706.6385 787.7512 254.5958 2278.3097 893.7869 336.8724
4 7977.788 400.7593 1505.9894 715.7045 1599.3564 343.6466 624.3485
5 8514.035 668.4213 952.6568 415.7571 2075.2624 688.6697 144.2359
6 6896.480 1488.4885 2602.7666 1907.2050 456.0669 996.3318 1643.4692
7 8978.509 1091.4266 511.7714 621.1159 2519.8371 1143.2913 481.7043
8 7229.322 1018.5534 2231.2810 1481.0287 839.7952 636.4621 1286.4875
9 9795.429 1746.6731 349.7070 1182.8871 3356.4636 1967.4115 1308.1430
10 9751.283 1695.3410 316.8296 1129.7752 3318.3164 1928.5321 1272.3671
11 9819.460 1729.3406 469.9372 1178.7001 3422.7249 2038.0182 1399.6214
12 2621.198 5779.6117 7181.1079 6352.6443 4378.6779 5634.8310 6279.0165
13 1810.597 7356.9224 8576.8230 7869.1950 5582.7210 6972.0480 7620.9851
14 9680.267 1611.9525 274.2754 1053.6555 3258.0298 1868.9296 1218.5885
15 0.000 8115.4794 9456.7048 8662.3895 6486.2148 7848.9759 8511.1082
16 8115.479 0.0000 1439.9021 585.0239 1852.9088 704.7432 764.0216
17 9456.705 1439.9021 0.0000 889.7365 3017.3596 1633.0753 974.5661
18 8662.389 585.0239 889.7365 0.0000 2310.1035 966.9902 543.3930
19 6486.215 1852.9088 3017.3596 2310.1035 0.0000 1393.4231 2050.4951
20 7848.976 704.7432 1633.0753 966.9902 1393.4231 0.0000 665.6561
21 8511.108 764.0216 974.5661 543.3930 2050.4951 665.6561 0.0000
22 8240.916 823.1926 1269.1611 801.6552 1769.8997 448.1752 339.1327
23 9074.561 1079.0857 389.0784 548.5864 2637.8334 1249.6051 599.1252
24 9897.377 1847.1784 447.7030 1282.6726 3457.1906 2068.6177 1408.4364
25 8530.027 797.1311 951.3601 578.5006 2071.1019 701.0357 115.6676
26 8091.347 712.4619 1396.4641 818.7179 1624.6954 258.7103 426.9930
27 6684.016 1542.1602 2793.9213 2026.4528 413.1816 1175.5250 1839.0179
28 8144.632 686.1968 1336.6555 751.3235 1685.1567 319.7077 377.8955
29 9499.249 1479.3951 114.5688 921.9539 3053.6734 1664.5255 1004.1668
30 9475.744 1428.9293 126.3922 868.5171 3043.7904 1653.5501 1000.5728
31 8956.035 989.4413 519.6710 475.0863 2514.6795 1123.8385 474.6989
32 8960.758 977.9131 507.8405 457.3215 2524.8614 1135.8860 491.3319
33 9750.863 1738.4378 303.4864 1178.1795 3300.3401 1917.1148 1253.5155
34 9768.995 1742.1185 316.1186 1179.5067 3323.1058 1937.6305 1275.3925
35 9563.585 1562.8215 133.6301 1006.9052 3113.8404 1730.4421 1067.5083
36 9739.879 1746.0624 307.0831 1191.2577 3285.2597 1906.4519 1242.0326
37 9814.211 1788.1320 359.6359 1227.3504 3367.1274 1982.5405 1319.6170
38 9720.619 1713.3339 278.7131 1154.4263 3267.8192 1884.6098 1220.4188
22 23 24 25 26 27 28
1 1690.9468 2582.8900 3393.1055 2010.1142 1574.9727 573.6942 1625.5433
2 3633.8522 4504.3843 5324.4662 3937.5866 3500.1391 2157.7859 3555.3171
3 634.9693 404.8531 1211.4710 362.7478 696.5572 2024.9731 631.1022
4 556.1672 1118.9254 1932.0461 676.0399 405.8891 1315.8693 390.8683
5 405.9372 567.6839 1388.3346 191.1361 473.0507 1843.3695 404.7159
6 1352.7572 2225.8140 3042.9009 1661.2122 1220.1188 405.8694 1269.5480
7 760.7943 210.7131 946.4280 454.0418 897.8953 2312.9440 839.6452
8 1047.0439 1847.2534 2670.7031 1310.4408 879.9472 581.7517 927.0690
9 1608.3709 721.9896 102.6596 1292.8476 1733.7263 3127.1142 1674.6322
10 1574.5142 681.5475 153.6847 1258.1506 1697.4024 3084.2965 1636.7452
11 1703.7726 801.5816 283.6917 1381.2541 1816.4204 3173.1683 1753.5116
12 6048.1084 6803.2633 7611.7037 6299.3364 5880.8204 4496.8592 5931.7737
13 7323.6509 8205.2841 9021.2198 7629.6377 7194.3996 5864.1362 7255.9648
14 1525.1488 621.9317 245.8190 1202.8421 1641.2322 3019.3044 1581.5293
15 8240.9161 9074.5613 9897.3769 8530.0274 8091.3469 6684.0163 8144.6320
16 823.1926 1079.0857 1847.1784 797.1311 712.4619 1542.1602 686.1968
17 1269.1611 389.0784 447.7030 951.3601 1396.4641 2793.9213 1336.6555
18 801.6552 548.5864 1282.6726 578.5006 818.7179 2026.4528 751.3235
19 1769.8997 2637.8334 3457.1906 2071.1019 1624.6954 413.1816 1685.1567
20 448.1752 1249.6051 2068.6177 701.0357 258.7103 1175.5250 319.7077
21 339.1327 599.1252 1408.4364 115.6676 426.9930 1839.0179 377.8955
22 0.0000 909.1293 1706.2725 334.9597 205.3047 1590.9324 168.9201
23 909.1293 0.0000 823.7560 586.7521 1020.0147 2407.4214 959.8885
24 1706.2725 823.7560 0.0000 1392.2651 1833.9722 3228.9704 1774.7507
25 334.9597 586.7521 1392.2651 0.0000 452.6433 1869.4464 404.2277
26 205.3047 1020.0147 1833.9722 452.6433 0.0000 1427.1300 104.5658
27 1590.9324 2407.4214 3228.9704 1869.4464 1427.1300 0.0000 1478.7968
28 168.9201 959.8885 1774.7507 404.2277 104.5658 1478.7968 0.0000
29 1307.7951 426.4692 405.5145 991.5271 1430.2622 2827.7026 1373.1009
30 1310.6353 407.9669 425.3834 989.8217 1425.0035 2808.8380 1365.8781
31 790.2658 140.0000 946.8865 476.2667 896.7999 2282.7356 835.7559
32 805.9950 123.2071 938.5489 489.7571 910.1154 2291.7535 847.5022
33 1540.3266 682.8309 189.7050 1231.9805 1676.7427 3082.9402 1617.6650
34 1566.5896 697.0072 148.9664 1255.7201 1699.5849 3101.0732 1639.8692
35 1355.5290 498.1957 355.5995 1045.8580 1490.4808 2895.8317 1430.9140
36 1521.8883 682.2199 240.0250 1216.6059 1662.8752 3074.5959 1604.6775
37 1609.5409 742.7907 121.0620 1298.5815 1743.4707 3147.0758 1684.6145
38 1507.9499 653.3215 219.4334 1199.6904 1643.8939 3051.6672 1585.7563
29 30 31 32 33 34 35
1 2994.08166 2989.57857 2459.74206 2471.55316 3228.78414 3254.21450 3043.5765
2 4924.29183 4911.66316 4385.17639 4392.67754 5165.53889 5188.63094 4979.0259
3 819.93048 790.00190 292.84467 293.99660 1080.12870 1091.27494 896.8238
4 1537.86020 1509.31872 997.42669 1001.47891 1798.46323 1811.21479 1613.5077
5 988.13612 972.14196 444.05743 451.32139 1241.08461 1258.75415 1054.1594
6 2642.24753 2632.74021 2102.20265 2112.42136 2884.06449 2907.04954 2697.5083
7 551.59405 562.95026 180.84800 195.44309 781.37827 806.69883 595.8800
8 2270.77366 2250.48972 1727.44812 1732.87651 2523.59208 2541.72382 2336.3240
9 305.31459 322.84362 845.46614 836.78731 136.07718 88.94942 266.0038
10 276.33856 279.77491 805.16893 795.31189 162.58844 118.19475 249.2188
11 463.54396 424.67870 929.93064 912.83241 375.72729 330.94410 450.0600
12 7225.09315 7188.55771 6693.88475 6692.94143 7482.92343 7494.63395 7298.7550
13 8620.27998 8612.24884 8087.15370 8096.07504 8858.01473 8883.55661 8672.9956
14 250.45359 223.22634 750.05533 736.91994 235.04680 201.24115 254.3344
15 9499.24892 9475.74435 8956.03450 8960.75767 9750.86304 9768.99478 9563.5848
16 1479.39515 1428.92932 989.44126 977.91308 1738.43780 1742.11854 1562.8215
17 114.56876 126.39225 519.67105 507.84053 303.48641 316.11865 133.6301
18 921.95390 868.51713 475.08631 457.32155 1178.17953 1179.50668 1006.9052
19 3053.67336 3043.79040 2514.67950 2524.86138 3300.34013 3323.10578 3113.8404
20 1664.52546 1653.55012 1123.83851 1135.88600 1917.11476 1937.63051 1730.4421
21 1004.16682 1000.57284 474.69885 491.33186 1253.51546 1275.39249 1067.5083
22 1307.79509 1310.63534 790.26578 805.99504 1540.32659 1566.58961 1355.5290
23 426.46923 407.96691 140.00000 123.20714 682.83087 697.00717 498.1957
24 405.51449 425.38336 946.88648 938.54888 189.70503 148.96644 355.5995
25 991.52711 989.82170 476.26673 489.75708 1231.98052 1255.72011 1045.8580
26 1430.26221 1425.00351 896.79987 910.11538 1676.74268 1699.58495 1490.4808
27 2827.70260 2808.83802 2282.73564 2291.75348 3082.94016 3101.07320 2895.8317
28 1373.10087 1365.87811 835.75595 847.50221 1617.66498 1639.86920 1430.9140
29 0.00000 96.73159 546.05494 540.84009 273.91605 283.37078 124.9600
30 96.73159 0.00000 532.90994 521.11515 323.07120 319.89061 191.8697
31 546.05494 532.90994 0.00000 53.47897 804.06219 819.72495 618.8239
32 540.84009 521.11515 53.47897 0.00000 798.16790 812.40938 613.2789
33 273.91605 323.07120 804.06219 798.16790 0.00000 49.78956 187.4433
34 283.37078 319.89061 819.72495 812.40938 49.78956 0.00000 211.3859
35 124.95999 191.86975 618.82388 613.27889 187.44332 211.38590 0.0000
36 289.31644 348.45516 802.68425 797.40454 63.92183 111.28791 186.9412
37 328.34433 365.50103 866.29672 858.96100 74.31016 50.44799 254.5172
38 243.73141 300.67258 773.30136 768.33782 39.17908 78.82893 158.5875
36 37 38
1 3210.31790 3297.95315 3197.15467
2 5148.48308 5232.31937 5134.34270
3 1082.68832 1138.66676 1051.73523
4 1797.68796 1858.54029 1768.96891
5 1234.54081 1304.81991 1210.32805
6 2868.91094 2951.68189 2852.59023
7 765.72972 849.56695 749.11348
8 2514.18357 2587.12949 2492.85499
9 196.55788 98.17841 155.60848
10 223.74763 145.49570 176.56160
11 427.32306 335.45044 398.62012
12 7478.73726 7539.62353 7454.60824
13 8837.25891 8925.53500 8825.83480
14 285.44877 231.05627 239.31151
15 9739.87885 9814.21062 9720.61906
16 1746.06243 1788.13199 1713.33388
17 307.08305 359.63593 278.71311
18 1191.25774 1227.35040 1154.42626
19 3285.25965 3367.12741 3267.81915
20 1906.45194 1982.54054 1884.60977
21 1242.03261 1319.61699 1220.41878
22 1521.88830 1609.54093 1507.94993
23 682.21991 742.79068 653.32151
24 240.02500 121.06197 219.43336
25 1216.60593 1298.58153 1199.69038
26 1662.87522 1743.47068 1643.89385
27 3074.59591 3147.07579 3051.66725
28 1604.67754 1684.61450 1585.75629
29 289.31644 328.34433 243.73141
30 348.45516 365.50103 300.67258
31 802.68425 866.29672 773.30136
32 797.40454 858.96100 768.33782
33 63.92183 74.31016 39.17908
34 111.28791 50.44799 78.82893
35 186.94117 254.51719 158.58752
36 0.00000 119.80818 63.61604
37 119.80818 0.00000 108.00463
38 63.61604 108.00463 0.00000Berdasarkan output matriks jarak (D) tersebut dapat dilihat bahwa Jawa Tengah, Jawa Timur, dan Jawa Barat memiliki jarak yang besar dengan provinsi lainnya, hal ini berkemungkinan disebabkan oleh jumlah desa sangat tinggi sehingga berbeda jauh dari provinsi lain.
4.2 Nilai Eigen dan Vektor Eigen dari Matriks B
Berikut nilai eigen dan vektor eigen berdasarkan output RStudio:
> A = D^2
> I = diag(38)
> J = matrix(rep(1,38), nrow=38, ncol=38)
> V = I-(1/38)*J
>
> aa = V %*% A
> BB = aa %*% V
> B = (-1/2) * BB
> eigen_result = eigen(B)
> eigenvalues = eigen_result$values
> eigenvalues
[1] 2.107616e+08 4.482196e+06 3.996089e+05 1.260848e+05 1.379357e+04
[6] 2.855166e-08 2.210561e-08 1.177276e-08 7.276424e-09 6.333031e-09
[11] 4.886929e-09 4.148171e-09 2.795739e-09 2.612993e-09 2.248778e-09
[16] 1.963071e-09 9.308295e-10 7.753821e-10 6.988404e-10 6.843666e-10
[21] 6.331280e-10 5.727939e-10 1.780134e-10 8.320569e-11 -2.501932e-10
[26] -4.094219e-10 -4.158579e-10 -6.688200e-10 -6.836167e-10 -8.298968e-10
[31] -1.316144e-09 -1.324366e-09 -1.375467e-09 -1.563279e-09 -1.851983e-09
[36] -3.245074e-09 -4.545197e-09 -1.901008e-08
> eigenvectors = eigen_result$vectors
> eigenvectors
[,1] [,2] [,3] [,4] [,5]
[1,] -0.101210785 0.2188611577 -0.242693569 0.307021397 -0.115416810
[2,] -0.236096960 0.1781591204 0.003524302 0.359971827 0.162796085
[3,] 0.046712208 -0.0666918441 -0.104607912 -0.019323054 -0.342701169
[4,] -0.002379258 -0.0968969198 -0.247214025 -0.110862276 0.019349505
[5,] 0.034070808 -0.0016077876 -0.080412370 -0.019157488 0.145830674
[6,] -0.078747467 0.1216207554 -0.225722541 0.158649353 0.038683222
[7,] 0.065478045 0.0685582678 0.053105777 0.035241933 0.097166336
[8,] -0.054385249 -0.0254517531 -0.085114436 -0.045616575 0.173199114
[9,] 0.122432455 -0.0222614328 0.066141148 -0.008843094 0.041605516
[10,] 0.119515201 -0.0402228883 0.039610947 0.045496416 -0.037701260
[11,] 0.124394725 -0.1331817024 0.154124511 0.276102838 -0.495140789
[12,] -0.386716955 -0.6417083821 0.380058002 -0.141384237 0.056711760
[13,] -0.486996599 0.5039750705 0.488357451 -0.173226130 -0.154579439
[14,] 0.114664165 -0.0710086605 0.114334644 0.032320265 -0.202589792
[15,] -0.551658652 -0.1851013480 -0.286935351 0.299091153 0.024507507
[16,] 0.007053917 -0.2556167982 0.014312842 0.001425159 0.053882256
[17,] 0.098912546 -0.0024771207 0.137871246 0.113724022 0.042175203
[18,] 0.044975953 -0.1804636979 -0.163683819 0.094680404 0.133857140
[19,] -0.107473210 0.1238139970 -0.136205277 -0.330890670 0.254645749
[20,] -0.012278007 0.0417612025 -0.128949372 -0.320909905 -0.071738840
[21,] 0.033316694 0.0540346352 -0.041064236 -0.239934919 0.090326802
[22,] 0.013979838 0.1299879222 -0.028107910 0.041794944 -0.050177353
[23,] 0.072718613 -0.0169907275 0.043024661 -0.018520051 -0.004323626
[24,] 0.129439934 -0.0176005450 0.070308729 0.017166191 0.041852188
[25,] 0.034424786 0.0694808409 0.094931058 -0.066386955 -0.159751237
[26,] 0.004112871 0.0777042221 -0.038180067 -0.228885758 0.034920691
[27,] -0.092101284 -0.0231144461 -0.338535380 -0.281476716 -0.482171680
[28,] 0.008040301 0.0665938344 -0.098766264 -0.020876161 0.157572551
[29,] 0.101876642 -0.0007173216 0.038789658 -0.127279134 0.010264192
[30,] 0.100497859 -0.0441788599 0.047466288 -0.072212323 0.085207766
[31,] 0.064546422 -0.0061808137 -0.061582875 -0.085552339 -0.070951567
[32,] 0.064945666 -0.0199355597 -0.024851403 -0.025900843 0.197034294
[33,] 0.119016058 0.0342294195 0.091424757 0.100153748 0.041961970
[34,] 0.120417989 0.0130510272 0.080028708 0.096151593 0.047840474
[35,] 0.106127693 0.0320413571 0.076631048 0.098440511 0.007775140
[36,] 0.118025391 0.0594902917 0.137114964 0.132505666 0.103846500
[37,] 0.123464683 0.0183695865 0.113461725 0.085328730 0.053303777
[38,] 0.116882961 0.0396759007 0.088004340 0.041972479 0.070927150
[,6] [,7] [,8] [,9] [,10]
[1,] 6.706042e-01 0.000000000 0.0000000000 0.000000000 0.000000000
[2,] -5.191448e-01 -0.140270975 -0.1514390629 -0.091754116 -0.425761933
[3,] 1.712818e-02 0.005316303 -0.0723694151 -0.101114507 -0.315465999
[4,] -2.875591e-02 -0.199767854 -0.1042589414 0.146926534 -0.047960236
[5,] 2.355258e-02 0.135866673 0.0476786993 0.034735187 -0.096172717
[6,] 5.132607e-02 -0.442669004 -0.1344218689 -0.142075143 0.228559583
[7,] -5.385286e-02 0.054461083 0.0767372558 0.142948013 0.313659281
[8,] -1.330452e-02 0.199737101 0.3883259899 -0.526920019 0.215911404
[9,] 1.692599e-01 -0.076123139 0.1723275709 -0.138390323 -0.307786508
[10,] -3.807678e-02 0.194367440 -0.3790703955 -0.063628139 0.061902151
[11,] 4.839157e-02 0.281773814 -0.0695496269 -0.281639387 -0.013114009
[12,] 1.799601e-01 -0.222787634 -0.1530311880 -0.044005051 0.064214854
[13,] 9.232815e-02 0.046096025 0.0812493057 -0.066481326 -0.021123220
[14,] -1.948681e-01 -0.072776031 0.4192254078 0.244412769 -0.024556398
[15,] 3.561415e-03 0.315256241 0.2149372432 0.363558435 -0.006428211
[16,] 5.796457e-02 0.117803925 -0.0253999296 -0.109952658 -0.081676229
[17,] 1.551030e-01 -0.163867909 0.1407772611 -0.039691418 -0.258737006
[18,] 1.147184e-02 0.029253249 0.0764904420 -0.187961670 -0.032566007
[19,] 2.134003e-01 0.132821294 -0.1884927583 -0.057217294 -0.272653314
[20,] -3.316532e-02 0.129593619 -0.0328931373 0.029659104 -0.109670067
[21,] -3.257859e-02 0.107541481 0.1179246690 0.050914399 0.076537208
[22,] -6.558768e-02 -0.134308662 -0.0906437030 -0.053659679 0.306169668
[23,] 3.835128e-02 -0.001996809 0.0724906136 0.140164412 -0.069433887
[24,] 6.090030e-02 0.111692593 -0.0126559242 0.045729647 -0.150417279
[25,] 3.571594e-05 -0.061392520 0.0642064228 0.116815244 0.072635203
[26,] -9.080849e-03 0.160615254 -0.0361851635 0.040773688 0.019384520
[27,] -7.909437e-02 -0.138579887 -0.0482208555 0.017688795 0.020364333
[28,] 6.973203e-02 0.021563694 -0.0917331395 -0.095726729 -0.039040498
[29,] 2.474477e-02 -0.122273361 0.0689997787 0.002994088 -0.085583103
[30,] 1.112251e-01 0.006178112 -0.0585934337 0.163232149 -0.069866892
[31,] -5.506870e-02 0.153100183 0.1991673073 0.102801557 -0.080137985
[32,] 1.040422e-02 -0.075655543 0.2073583973 -0.071493680 0.059745181
[33,] 2.930157e-02 0.015518932 0.0111901777 -0.025007701 -0.115646803
[34,] 9.089867e-02 0.130900096 0.0779409372 0.109071073 -0.096414748
[35,] 9.456640e-02 -0.235921813 0.2015922209 0.029317807 0.079350257
[36,] -5.787103e-02 0.284254645 -0.2848658368 0.030959098 0.200207183
[37,] 1.057247e-01 0.066836803 -0.1680811192 0.376510783 0.082288285
[38,] 1.319421e-01 -0.184130474 -0.0008417089 0.189562596 -0.113013901
[,11] [,12] [,13] [,14] [,15]
[1,] 0.000000000 0.000000000 0.000000000 0.000000000 0.00000000
[2,] -0.143661879 -0.056659740 -0.074450920 0.119484531 0.08556318
[3,] -0.076599095 0.518487849 -0.004868837 -0.033955892 -0.11274198
[4,] 0.134444460 0.050570671 -0.043830682 -0.702363570 0.24956681
[5,] -0.153219606 0.502733248 0.126142754 -0.148811264 -0.09092149
[6,] 0.175626417 -0.012575785 0.064531299 0.066827336 -0.06535942
[7,] -0.697088981 -0.021799724 0.064960063 -0.149746958 -0.22522412
[8,] 0.042931799 -0.002367884 0.263699826 0.012981326 0.24701581
[9,] -0.137612897 -0.412311186 -0.195290052 -0.310641188 -0.19799032
[10,] 0.055147559 -0.100394012 0.055274232 -0.214130776 0.02948680
[11,] -0.176022528 -0.092190301 -0.124666030 -0.073004790 0.21950361
[12,] -0.088561210 0.033402262 -0.022605107 0.021433659 -0.05969318
[13,] 0.181830723 0.102662826 -0.015934696 -0.168910668 -0.01689737
[14,] 0.212475502 0.077229848 0.153917220 -0.086455444 0.12987296
[15,] -0.051353423 -0.052878237 -0.048175799 -0.052660045 0.08233222
[16,] 0.132159200 -0.009730043 0.032420617 0.095909418 0.15614392
[17,] -0.021567932 -0.070027993 0.142780053 -0.035423891 0.01992145
[18,] 0.173712364 0.119398618 -0.167450192 -0.015859206 -0.19539648
[19,] -0.222626748 0.050537489 -0.067457520 0.076930537 0.23452827
[20,] 0.015014836 -0.063409450 0.207810219 0.025473022 -0.16687059
[21,] 0.047119172 0.005302495 -0.378069821 -0.034867624 0.04214360
[22,] -0.121479969 0.062757558 -0.013241480 0.005854239 0.18829979
[23,] -0.057295717 0.002837490 0.182895467 0.055607716 0.20197698
[24,] -0.010241144 0.005178499 -0.200692791 0.219087994 0.26062606
[25,] -0.204391715 0.018713627 -0.049052541 0.021218374 0.11037863
[26,] 0.030361951 -0.109452950 -0.257953258 0.067820551 -0.00982438
[27,] -0.043306564 -0.148527572 -0.106898601 0.205596952 -0.03272304
[28,] -0.039832385 -0.025364031 0.216375851 0.003715202 0.08105703
[29,] -0.128483095 0.039678423 0.123062278 -0.046098879 0.41991350
[30,] 0.059811145 0.073733370 0.224167202 0.232902464 -0.15479111
[31,] 0.101468324 -0.110122861 0.066853715 -0.018590792 -0.18119117
[32,] 0.038359627 0.065702644 -0.437628856 0.014512229 0.05234197
[33,] 0.028548816 0.186473558 -0.097503118 -0.084571484 -0.18104699
[34,] 0.095341517 -0.107830705 0.029544766 0.113466837 -0.14510574
[35,] -0.071387494 0.270727544 -0.278326335 0.115398440 0.02855960
[36,] 0.219135935 0.059867052 -0.052820829 -0.160030480 -0.07169901
[37,] 0.111185839 0.058404849 -0.085968618 0.141439706 0.25713085
[38,] 0.003649745 -0.219135421 0.120468960 -0.047996150 0.06490097
[,16] [,17] [,18] [,19] [,20]
[1,] 0.000000000 0.00000000 0.000000000 0.000000000 0.000000000
[2,] -0.028965787 0.00799587 0.030258429 0.050983843 0.020358271
[3,] -0.079718276 -0.11728674 0.087396242 0.024873415 0.058524289
[4,] -0.012297173 0.09509920 -0.020076187 0.101562663 0.045472304
[5,] 0.163498444 0.13012675 0.042601022 0.077690335 -0.185242460
[6,] 0.021373196 -0.16057126 -0.015721508 0.001668017 -0.056918204
[7,] -0.049653858 0.05056895 0.049043602 -0.061154958 0.036196540
[8,] -0.217496765 0.14756885 0.033584397 0.126877767 -0.040701772
[9,] 0.051753885 0.24844882 -0.013523745 0.251268034 -0.046455322
[10,] -0.038378772 0.02562606 0.050442749 0.024400721 0.135055120
[11,] 0.009652313 -0.11078701 0.069495150 -0.022677417 -0.081431439
[12,] -0.037045537 -0.08068321 0.004892784 0.049626879 -0.037046639
[13,] -0.005667942 0.05956859 0.005319127 0.011703364 -0.005927348
[14,] -0.138707142 0.13285273 0.126705109 -0.004770688 -0.044138287
[15,] 0.039775592 -0.04030062 -0.059288387 0.016180551 -0.002486631
[16,] 0.302874767 0.18725186 -0.159089131 -0.132299873 0.110034719
[17,] 0.102192668 -0.33358624 -0.069563244 -0.010392231 0.009578701
[18,] -0.358937780 -0.17777221 0.070434619 -0.228036077 -0.111971273
[19,] -0.202496066 0.08652055 0.043190688 -0.271416129 -0.018043402
[20,] -0.132295240 -0.09317912 0.113670763 -0.023147024 -0.011649300
[21,] 0.188279550 -0.36703622 0.100140104 -0.068418327 0.009073962
[22,] 0.234661870 0.01373517 -0.146989142 -0.055330890 -0.421215007
[23,] 0.023746018 -0.33170001 0.334138891 0.256782716 -0.083664429
[24,] -0.149635259 0.18290500 -0.259238110 0.104626236 -0.066916102
[25,] -0.390537343 -0.25552649 -0.295903992 0.169150055 0.132026178
[26,] 0.168337914 -0.13031234 0.065515305 0.120693134 -0.086359678
[27,] -0.011981640 0.23374289 0.038830049 0.125487890 -0.026218560
[28,] 0.042400939 -0.21705410 -0.100424144 0.437215402 0.360366679
[29,] 0.298340965 -0.09800514 -0.089198349 -0.247913942 0.019880474
[30,] 0.048751514 0.20005717 -0.142546118 0.283551866 -0.089256502
[31,] 0.059131806 -0.19961967 -0.542243283 -0.199833215 0.113561951
[32,] -0.077607274 -0.06706355 0.079983151 0.248429903 -0.099015803
[33,] 0.030958610 -0.02208019 -0.300714413 0.103047824 -0.386811306
[34,] 0.235678788 0.01961026 0.404488275 -0.060660608 0.012162596
[35,] 0.060571199 0.22353033 0.041682500 -0.112728302 0.554775175
[36,] -0.096944169 0.01042029 -0.026460471 -0.027835254 0.173733636
[37,] -0.175231239 0.05987052 0.042146883 0.174107427 -0.108637351
[38,] -0.319777505 0.03273320 0.086531765 -0.336643486 -0.151543888
[,21] [,22] [,23] [,24] [,25]
[1,] 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000
[2,] -0.038334322 0.031789032 -0.010121450 0.014574583 -0.067154129
[3,] -0.076631882 0.068260815 0.008311752 -0.219694752 0.172463617
[4,] 0.035917742 0.004862741 -0.098024392 -0.179276582 0.081035434
[5,] -0.100267393 -0.041226688 0.093778447 0.329827732 -0.199052044
[6,] -0.125701206 0.007475019 -0.082251146 0.023859594 0.051154766
[7,] -0.010146340 0.070373565 -0.022458074 -0.051839872 0.269626112
[8,] 0.094205645 0.090607365 0.088243715 -0.207365322 -0.110012125
[9,] -0.068877285 -0.027438275 -0.032196100 0.030149821 0.045285664
[10,] -0.049536633 0.154930623 0.120227382 0.080267089 -0.409190281
[11,] -0.225798320 0.007659173 -0.033828244 0.098690055 -0.115550500
[12,] -0.073388019 -0.046915610 0.106893829 -0.070968977 -0.032420325
[13,] -0.026290642 0.017678232 -0.041631771 0.038536911 0.049653520
[14,] -0.291108736 0.067140997 0.051464917 0.210097267 0.194890416
[15,] 0.066891526 0.001029833 -0.035324966 -0.001522310 -0.128421755
[16,] 0.220158403 0.213749441 -0.233247143 0.487259386 0.319872251
[17,] 0.117382509 0.031740128 0.041980766 0.077549456 0.131976752
[18,] -0.103406864 -0.058724922 -0.110756772 0.046633813 0.113977801
[19,] -0.199217311 0.024318320 -0.102621667 0.089048934 0.100517353
[20,] 0.243751950 -0.243039448 0.071541817 0.138262763 -0.112305212
[21,] 0.050235169 0.383488799 0.288630188 0.035838131 -0.016814583
[22,] -0.140760456 0.007876678 -0.016383201 0.077958482 0.093484534
[23,] 0.193182256 0.273383464 -0.376556882 -0.136192619 -0.007320142
[24,] 0.106037220 0.003340970 0.087083996 -0.325760439 0.242580896
[25,] 0.176947670 -0.158143784 -0.257378425 0.272559279 -0.088060028
[26,] -0.214749317 0.130247296 -0.210058807 -0.114912163 -0.038101862
[27,] 0.170129091 0.007693718 0.075446410 0.115805039 -0.014448250
[28,] -0.187715785 -0.003400222 0.340736711 0.208408832 0.156145321
[29,] 0.043100522 -0.413443768 -0.056357810 -0.135439251 -0.232599895
[30,] -0.266261100 0.180619969 -0.349973993 -0.099110077 -0.254808745
[31,] -0.316728844 -0.022818158 0.014119773 -0.143866101 -0.073787346
[32,] -0.116256911 -0.297942780 -0.152322743 0.254919886 -0.166129814
[33,] 0.441653839 0.092810752 0.167471023 -0.020059125 -0.137439485
[34,] 0.039837256 -0.363928222 0.046930801 -0.085966127 0.063252690
[35,] 0.079865255 0.150772843 -0.008532584 -0.051983052 -0.272784520
[36,] 0.146222883 -0.122774149 -0.274752849 -0.001384784 0.137325427
[37,] -0.086308363 -0.118869574 0.346237416 0.016168130 0.095609292
[38,] 0.008056758 0.296619722 0.093416405 0.151493809 -0.238272979
[,26] [,27] [,28] [,29] [,30]
[1,] 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000
[2,] 0.095365294 -0.009752954 0.098522719 0.143399733 0.051393359
[3,] 0.138814826 -0.176586119 -0.218411227 -0.080047733 0.018351456
[4,] 0.029576402 -0.007628420 0.132991205 0.268168049 0.083564102
[5,] -0.265637926 -0.120047987 -0.041890162 0.279664502 -0.067541656
[6,] 0.160138218 0.134044924 0.102049306 0.168461489 0.014090119
[7,] 0.112656760 0.030151538 0.216694810 0.166650152 -0.116429989
[8,] 0.209294163 -0.120000103 -0.066746410 0.143943586 0.001736586
[9,] 0.006676837 0.109640688 -0.103960647 -0.171863357 0.127690241
[10,] 0.293273641 -0.085771327 -0.002221225 0.054633661 0.033860679
[11,] -0.065111891 0.172937807 0.206543444 0.114861791 0.014611343
[12,] 0.092314121 -0.093264767 0.094841476 0.030940681 0.019402172
[13,] -0.060204466 0.006538397 0.188300055 0.132981801 -0.006111398
[14,] 0.245870396 0.002157276 0.123641324 -0.271073188 -0.028459726
[15,] 0.079688059 -0.072901081 0.007283024 -0.143560648 -0.002417219
[16,] 0.052192407 0.087016484 0.037621332 0.195569141 -0.007931611
[17,] 0.308880655 -0.219614759 -0.159027885 0.210382745 -0.253419910
[18,] -0.152830795 0.054615059 0.399275088 -0.006260237 0.013857751
[19,] 0.345458324 0.080376563 0.040055267 -0.168168726 0.100840376
[20,] 0.097038779 0.290231902 0.147653211 0.130230564 0.008673079
[21,] -0.037104650 -0.018365548 -0.014014829 0.008044184 0.386896912
[22,] 0.154461197 -0.208510237 0.034635839 -0.119641711 0.265713071
[23,] -0.071197191 0.169334224 0.174354381 0.028359842 0.025382406
[24,] -0.238823103 -0.179756471 0.174310437 0.204580142 0.054095585
[25,] 0.045503987 -0.146784086 -0.176996327 0.047410652 0.285276954
[26,] 0.112641290 -0.164407293 0.012526642 0.021115829 -0.548878909
[27,] 0.071261628 -0.271662391 0.240803175 0.081988493 -0.116004976
[28,] -0.144444741 -0.159212430 0.357493937 -0.240926760 -0.033077797
[29,] -0.096673965 0.012031749 0.183125263 -0.220493790 -0.116566797
[30,] 0.045901471 -0.004055396 0.144457082 0.006310691 0.244804940
[31,] 0.116583307 0.033700992 0.100276403 0.290645318 0.025269049
[32,] 0.061163725 -0.082749184 -0.026916020 0.100821279 -0.112306809
[33,] 0.286784993 0.100013923 0.317951021 -0.245175400 -0.093157088
[34,] 0.223011824 -0.331229045 0.190152670 0.210997810 0.358064804
[35,] 0.157029083 0.088272408 0.192181492 0.012688870 -0.024674144
[36,] 0.076548719 -0.284979371 0.071752215 -0.194119753 -0.075984209
[37,] 0.186299020 0.318430139 -0.094306173 0.218673855 -0.120053178
[38,] -0.211036797 -0.370623742 0.075261876 0.041338255 -0.112190010
[,31] [,32] [,33] [,34] [,35]
[1,] 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000
[2,] 0.159368314 -0.020421823 -0.026487448 0.015377462 -0.053347620
[3,] 0.068334445 -0.046687046 0.241825012 -0.188293300 -0.047938514
[4,] -0.006574128 0.097645156 -0.002309441 0.257564542 -0.080923896
[5,] 0.186904162 0.013325643 -0.175952528 -0.164013864 0.080775608
[6,] 0.164059236 -0.113556042 0.070612680 -0.431119073 -0.051778301
[7,] -0.099121296 -0.168470543 0.030179215 -0.086094093 -0.064150487
[8,] 0.198293646 -0.022145822 -0.046318220 -0.009805890 -0.043889663
[9,] 0.243455257 -0.020382702 -0.036935504 -0.308436914 -0.029911239
[10,] -0.460034299 0.002752785 -0.057947905 -0.384733077 0.083897129
[11,] 0.136713078 -0.045031159 0.067110142 0.213519622 -0.105227624
[12,] 0.144310513 0.021984962 0.037746460 -0.043320741 0.046903932
[13,] -0.103436732 -0.063700298 -0.025943008 -0.066299802 -0.003987451
[14,] -0.064403701 -0.008509369 0.006477295 -0.163289423 -0.039432228
[15,] -0.033589625 -0.038847624 0.083834279 -0.103823841 -0.117411110
[16,] 0.002305838 0.047977825 0.077815002 -0.128160721 -0.027457689
[17,] -0.197901810 -0.223343352 -0.132710874 0.210527043 -0.232134400
[18,] -0.157365904 0.067259119 -0.345721895 -0.053128273 -0.151334481
[19,] 0.013511334 -0.036461667 -0.053869778 0.107205418 0.192735977
[20,] 0.077948825 0.104874492 0.459442236 -0.046115741 -0.373536824
[21,] 0.126974917 -0.278964954 -0.059466476 -0.063712263 -0.206941202
[22,] 0.006107070 0.256139205 0.206442824 0.068480627 -0.057287517
[23,] 0.043748352 0.016534328 0.037884395 -0.114413489 0.417829850
[24,] -0.229740647 0.008035639 0.192634912 -0.308169205 -0.093475676
[25,] 0.080037661 0.258421051 -0.248130383 -0.113949614 -0.090245829
[26,] 0.131113384 0.416952126 -0.092579944 -0.109754474 -0.237821440
[27,] 0.069184120 -0.274500707 -0.235371604 0.043216476 0.155773648
[28,] 0.015402570 0.106790204 0.088345136 0.082601948 0.020959592
[29,] 0.039438886 -0.232174856 -0.138249182 -0.258560260 -0.138192576
[30,] -0.054789440 -0.261366209 -0.049081097 0.179100967 -0.362445233
[31,] 0.166540654 -0.012614584 0.096564597 -0.012415486 0.386032587
[32,] -0.227554105 -0.259750894 0.400384437 0.093295405 0.193868267
[33,] 0.030654649 0.085350878 -0.032126317 0.071678722 0.123903162
[34,] 0.032293793 0.224071066 -0.081905936 0.005818245 0.083685354
[35,] 0.104599955 0.194824071 0.072593906 0.063829576 -0.055074226
[36,] 0.406178969 -0.326119200 0.077690420 -0.008828202 -0.004764463
[37,] 0.244206408 -0.070720897 -0.170751860 -0.019852368 -0.030239290
[38,] 0.203021517 0.022596185 0.262371127 0.003715448 -0.006986835
[,36] [,37] [,38]
[1,] 0.00000000 0.000000000 0.570668753
[2,] -0.02583305 0.055645246 0.340615611
[3,] -0.29001943 0.200467009 -0.089668042
[4,] -0.07435931 -0.072223824 0.028953909
[5,] 0.27791458 -0.123263300 -0.015414785
[6,] -0.01200241 -0.331773986 -0.294449430
[7,] -0.17881736 0.007924405 0.071879397
[8,] -0.09524852 0.058874747 0.039123779
[9,] 0.11359784 0.207945027 -0.127348306
[10,] 0.03895523 0.116579554 0.066111015
[11,] 0.02144216 -0.216813055 -0.166866022
[12,] 0.08456493 -0.036757488 0.215211084
[13,] -0.07634689 0.120935815 -0.118529789
[14,] 0.11953008 -0.226297571 0.266824924
[15,] 0.02697930 0.043730632 -0.309017760
[16,] -0.28697662 0.111722562 0.047386917
[17,] 0.33453873 0.063246658 -0.157792158
[18,] 0.03771066 0.351077287 -0.029770465
[19,] 0.11775735 -0.197783926 -0.145720040
[20,] 0.21973525 0.066670356 0.124081414
[21,] -0.02450855 -0.076353638 0.153359761
[22,] 0.25389094 0.411874575 -0.014887591
[23,] 0.16814859 0.166110419 0.001732754
[24,] 0.24927800 -0.189206596 -0.012728350
[25,] -0.10047522 -0.099769607 0.023195498
[26,] -0.07807114 -0.083561785 0.095566280
[27,] 0.07735576 0.108099286 -0.004579665
[28,] -0.09148680 0.044560772 -0.104960393
[29,] -0.18275378 0.103223756 0.076314224
[30,] -0.12517978 0.019000757 -0.019665736
[31,] 0.02879987 0.122028322 0.084018023
[32,] -0.16508024 0.035914639 0.050153634
[33,] -0.17673397 -0.192193962 -0.032967574
[34,] -0.17694215 -0.117787917 -0.098485058
[35,] 0.21622216 0.118925419 -0.123392182
[36,] 0.23045222 0.013396919 0.074148729
[37,] -0.13394343 0.318002927 -0.096260772
[38,] -0.21686172 0.059458094 -0.1203444144.3 Keragaman Kumulatif
> cumulative_variance = cumsum(eigenvalues) / sum(eigenvalues)
> cumulative_variance
[1] 0.9767281 0.9974999 0.9993518 0.9999361 1.0000000 1.0000000 1.0000000
[8] 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000
[15] 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000
[22] 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000
[29] 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000
[36] 1.0000000 1.0000000 1.0000000Berdasarkan output tersebut, dapat dilihat bahwa dimensi 1 mampu menjelaskan 97,67% informasi jarak dan dimensi 1 + 2 mampu menjelaskan 99,75% informasi jarak. Hal ini menunjukkan bahwa model MDS sangat mampu merangkum informasi dari data asli cukup menggunakan dua dimensi saja. Dengan demikian, visualisasi dua dimensi sudah sangat representatif untuk menggambarkan hubungan antarprovinsi.
4.4 Koordinat MDS
> fit = cmdscale(D, k=2)
> fit
[,1] [,2]
1 -1469.34063 463.355261
2 -3427.56809 377.184177
3 678.15051 -141.194615
4 -34.54118 -205.142374
5 494.62735 -3.403879
6 -1143.22652 257.485693
7 950.58597 145.146057
8 -789.54486 -53.884407
9 1777.42897 -47.130117
10 1735.07735 -85.156668
11 1805.91648 -281.961601
12 -5614.21329 -1358.573435
13 -7070.03596 1066.975533
14 1664.65182 -150.333832
15 -8008.77566 -391.881704
16 102.40614 -541.171350
17 1435.97564 -5.244361
18 652.94420 -382.063244
19 -1560.25620 262.129048
20 -178.24755 88.413463
21 483.67941 114.397789
22 202.95410 275.199986
23 1055.70185 -35.971403
24 1879.16094 -37.262460
25 499.76627 147.099255
26 59.70914 164.509137
27 -1337.09228 -48.936048
28 116.72611 140.987116
29 1479.00730 -1.518656
30 1458.99064 -93.531933
31 937.06101 -13.085522
32 942.85709 -42.205965
33 1727.83098 72.467777
34 1748.18369 27.630586
35 1540.72247 67.835387
36 1713.44885 125.948067
37 1792.41447 38.890613
38 1696.86346 83.998630
>
> mds_df = data.frame(
+ Dim1 = fit[,1],
+ Dim2 = fit[,2],
+ Provinsi = data$Provinsi,
+ ID = 1:nrow(data))Koordinat hasil MDS menunjukkan posisi masing-masing provinsi dalam ruang dua dimensi. Posisi tersebut menggambarkan jarak psikologis atau kemiripan pola fasilitas pendidikan antarprovinsi.
Koordinat tersebut merepresentasikan:
- Provinsi yang berdekatan -> memiliki pola fasilitas pendidikan yang mirip
- Provinsi yang berjauhan -> memiliki pola fasilitas pendidikan yang berbeda
4.5 Visualisasi Plot
Visualisasi MDS disajikan dalam bentuk scatter plot dua dimensi dengan nama provinsi sebagai label. Setiap titik mewakili satu provinsi, dan kedekatan titik menggambarkan kemiripan pola fasilitas pendidikan.
> ggplot(mds_df, aes(x = Dim1, y = Dim2, color = Provinsi)) +
+ geom_point(size = 3) +
+ geom_text_repel(aes(label = Provinsi), size = 3, max.overlaps = 100) +
+ geom_text_repel(aes(label = ID), size = 3, color = "black",
+ box.padding = 0.3, max.overlaps = 100) +
+ theme_minimal() +
+ labs(
+ title = "Visualisasi MDS Jumlah Desa dengan Fasilitas Pendidikan",
+ x = "Dimensi 1",
+ y = "Dimensi 2",
+ color = "Provinsi") +
+ theme(
+ plot.title = element_text(hjust = 0.5, face = "bold"),
+ legend.position = "none")
4.5.1 Interpretasi Sumbu
Sumbu dalam MDS adalah dimensi laten yang diciptakan untuk meminimalkan stress (ketidaksesuaian) dan merepresentasikan struktur data asli.
Dimensi 1 (sumbu horizontal) kemungkinan besar merepresentasikan skala absolut, kepadatan, atau kompleksitas jaringan fasilitas pendidikan. Provinsi di Jawa memiliki jumlah desa yang sangat besar dan kepadatan penduduk yang tinggi, yang secara fundamental membedakan struktur distribusi fasilitas pendidikan mereka dari provinsi lain. Oleh karena itu, MDS memisahkan Jawa sebagai kelompok yang memiliki skala jauh lebih besar.
Dimensi 2 (sumbu vertikal) kemungkinan besar menjelaskan perbedaan jenis atau pola distribusi internal fasilitas di Jawa. Misalnya, ini mungkin membedakan provinsi yang distribusi fasilitasnya cenderung merata (Jawa Tengah) dari provinsi yang memiliki polarisasi tinggi antara wilayah padat dan wilayah yang jarang penduduk (Jawa Barat).
4.5.2 Analisi Pengelompokkan
Plot MDS ini membentuk tiga klaster utama yang menunjukkan kemiripan struktural:
- Klaster A
- Provinsi: Jawa Tengah, Jawa Timur, dan Jawa Barat.
- Karakteristik: Meskipun berada dalam satu klaster skala besar (sisi kiri plot), terdapat variasi signifikan di antara mereka (dipisahkan oleh Dimensi 2). Hal ini menggarisbawahi bahwa masalah atau pola distribusi fasilitas di Jawa harus dianalisis terpisah dari seluruh Indonesia.
- Klaster B
- Provinsi: Aceh, Sumatera Utara, Sumatera Selatan, dan Nusa Tenggara Timur.
- Karakteristik: Kelompok ini sedikit tersebar di sekitar pusat plot. Kelompok ini sangat rapat di sekitar pusat plot (Dimensi 1 dan 2 mendekati nol). Ini mungkin menunjukkan bahwa faktor geografis atau tantangan infrastruktur menyebabkan pola distribusi fasilitas yang berbeda, meskipun skalanya tetap lebih kecil daripada Jawa.
- Klaster C
- Provinsi: DKI Jakarta, Banten, DIY Yogyakarta, Bengkulu, Jambi, Kepulauan Bangka Belitung, Kepulauan Riau, juga mayoritas provinsi di Kepulauan Nusa Tenggara, Kalimantan, Maluku, Sulawesi, dan Papua
- Karakteristik: Kelompok ini sangat rapat di di sekitar sisi kanan Klaster B (Dimensi 1 dan 2 mendekati nol). Ini menunjukkan bahwa, setelah dikompensasi untuk skala yang lebih kecil daripada Jawa, provinsi-provinsi ini memiliki model distribusi fasilitas pendidikan yang sangat seragam.
Secara keseluruhan visualisasi MDS ini menunjukkan bahwa faktor skala dan kepadatan (diwakili oleh Pulau Jawa) adalah perbedaan terbesar dalam data (Dimensi 1). Setelah faktor skala dikendalikan, mayoritas provinsi di luar Jawa memiliki struktur distribusi fasilitas pendidikan yang sangat homogen (rapat di pusat plot).
4.6 Disparites
Matriks disparity dihitung berdasarkan jarak antar koordinat MDS (jarak hasil pemetaan ke ruang dua dimensi). Berikut matriks disparity berdasarkan output RStudio:
> disparities = matrix(0, nrow = 38, ncol = 38)
> for (i in 1:38) {
+ for (j in 1:38) {
+ disparities[i, j] = sqrt(sum((fit[i,] - fit[j,])^2))
+ }
+ }
> disparities
[,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] 0.0000 1960.123 2230.9636 1582.8893 2018.6714 385.6588 2440.7585
[2,] 1960.1225 0.000 4138.3139 3442.6350 3940.6173 2287.4755 4384.2986
[3,] 2230.9636 4138.314 0.0000 715.5549 229.4930 1864.5000 395.2367
[4,] 1582.8893 3442.635 715.5549 0.0000 566.3195 1201.3359 1045.5513
[5,] 2018.6714 3940.617 229.4930 566.3195 0.0000 1658.5019 479.5470
[6,] 385.6588 2287.475 1864.5000 1201.3359 1658.5019 0.0000 2096.8240
[7,] 2440.7585 4384.299 395.2367 1045.5513 479.5470 2096.8240 0.0000
[8,] 854.2009 2673.011 1470.2900 770.0062 1285.1640 471.2134 1751.4761
[9,] 3286.6561 5222.264 1103.2956 1818.8468 1283.5466 2936.4978 848.9049
[10,] 3251.0244 5183.307 1058.4114 1773.6816 1243.1411 2898.6267 817.5977
[11,] 3358.9889 5274.830 1136.5172 1842.0601 1340.5497 2998.0740 956.0393
[12,] 4527.6256 2791.822 6409.0447 5697.6437 6257.3492 4754.0898 6734.8171
[13,] 5633.1293 3707.207 7841.8154 7149.5784 7640.0159 5981.8345 8073.4222
[14,] 3193.5126 5119.471 986.5436 1700.0767 1179.2140 2837.3399 772.7861
[15,] 6595.1225 4645.312 8690.5426 7976.4207 8512.2722 6896.1905 8975.4420
[16,] 1865.3315 3647.478 701.0442 362.8637 665.6060 1479.6803 1091.0731
[17,] 2942.8639 4878.556 769.9230 1484.0414 941.3501 2592.5491 508.1539
[18,] 2284.4748 4150.547 242.1839 709.8853 410.4231 1906.6337 605.4257
[19,] 220.8113 1870.853 2274.4526 1595.6657 2071.9687 417.0555 2513.5659
[20,] 1344.4340 3262.127 886.6440 326.8433 679.1105 979.6784 1130.2582
[21,] 1983.9502 3920.066 321.1643 608.8173 118.3093 1633.1862 467.9179
[22,] 1682.8464 3631.954 631.8197 535.8477 403.3527 1346.2972 758.8593
[23,] 2573.9399 4502.267 391.9400 1103.2899 562.0189 2218.4235 209.4108
[24,] 3385.7172 5322.888 1205.4990 1921.0517 1384.9475 3036.7256 946.3215
[25,] 1994.3420 3934.068 339.0196 639.9677 150.5908 1646.6968 450.8239
[26,] 1557.9802 3493.756 689.8728 381.4779 466.2066 1206.5234 891.0872
[27,] 529.0860 2133.464 2017.3535 1311.8841 1832.2855 362.5993 2295.8963
[28,] 1618.4959 3552.156 628.3501 377.7399 404.5468 1265.3270 833.8702
[29,] 2984.7719 4921.168 812.9459 1527.1842 984.3818 2634.9940 548.3973
[30,] 2980.8132 4909.178 782.2934 1497.6963 968.5658 2625.7851 561.6427
[31,] 2453.1133 4382.043 288.8712 990.4023 442.5396 2097.8096 158.8086
[32,] 2464.6075 4390.502 282.6098 990.8863 449.9061 2107.5009 187.5114
[33,] 3220.9780 5164.397 1071.2053 1784.1029 1235.5354 2877.0128 780.6356
[34,] 3246.8937 5187.542 1083.2696 1797.8574 1253.9404 2900.5321 806.2084
[35,] 3035.9374 4977.912 887.5381 1598.7409 1048.5180 2690.6411 595.1790
[36,] 3200.6238 5147.152 1069.2090 1779.0700 1225.6663 2859.7021 763.1044
[37,] 3289.2577 5230.933 1128.7227 1843.1818 1298.4761 2943.7683 848.5078
[38,] 3188.8493 5132.812 1043.3063 1755.3816 1205.4090 2845.3838 748.7784
[,8] [,9] [,10] [,11] [,12] [,13] [,14]
[1,] 854.2009 3286.65607 3251.02444 3358.9889 4527.626 5633.129 3193.51263
[2,] 2673.0108 5222.26359 5183.30657 5274.8302 2791.822 3707.207 5119.47056
[3,] 1470.2900 1103.29564 1058.41136 1136.5172 6409.045 7841.815 986.54364
[4,] 770.0062 1818.84681 1773.68157 1842.0601 5697.644 7149.578 1700.07671
[5,] 1285.1640 1283.54664 1243.14107 1340.5497 6257.349 7640.016 1179.21400
[6,] 471.2134 2936.49779 2898.62674 2998.0740 4754.090 5981.834 2837.33987
[7,] 1751.4761 848.90487 817.59775 956.0393 6734.817 8073.422 772.78613
[8,] 0.0000 2566.98272 2524.81589 2605.4633 4997.963 6379.725 2456.09117
[9,] 2566.9827 0.00000 56.91817 236.5531 7507.081 8917.335 152.87149
[10,] 2524.8159 56.91817 0.00000 209.1659 7458.798 8880.171 95.95738
[11,] 2605.4633 236.55309 209.16588 0.0000 7497.828 8977.871 193.08437
[12,] 4997.9635 7507.08059 7458.79771 7497.8276 0.000 2828.906 7378.46327
[13,] 6379.7253 8917.33520 8880.17056 8977.8707 2828.906 0.000 8819.10498
[14,] 2456.0912 152.87149 95.95738 193.0844 7378.463 8819.105 0.00000
[15,] 7227.1388 9792.27526 9748.67949 9815.3076 2582.329 1734.790 9676.44276
[16,] 1016.3785 1746.36142 1695.15919 1723.1184 5774.763 7350.514 1610.39297
[17,] 2226.0520 344.01278 309.59297 461.9834 7178.904 8573.324 270.82051
[18,] 1479.3499 1173.30567 1122.12552 1157.3096 6342.778 7857.744 1037.90694
[19,] 832.9829 3351.98203 3313.58274 3409.8612 4365.919 5568.254 3251.17784
[20,] 627.6409 1960.36800 1921.18160 2018.4362 5625.255 6960.915 1858.29985
[21,] 1284.2971 1303.79415 1267.20906 1380.3665 6273.272 7613.542 1210.28040
[22,] 1045.6341 1607.13030 1573.93093 1697.0319 6042.239 7315.962 1522.37947
[23,] 1845.3337 721.81338 681.15363 789.5145 6799.783 8200.250 619.59570
[24,] 2668.7576 102.20941 151.83522 255.4260 7608.976 9017.065 242.48567
[25,] 1304.8822 1292.34168 1256.95516 1374.8169 6296.650 7625.489 1202.25820
[26,] 876.8855 1730.70870 1693.86885 1802.3807 5874.792 7186.634 1635.53267
[27,] 547.5698 3114.52177 3072.38314 3151.6353 4473.132 5840.540 3003.45620
[28,] 926.9854 1671.32346 1634.07520 1741.3356 5923.879 7246.172 1575.10060
[29,] 2269.1565 301.88723 269.38298 430.7178 7221.868 8615.557 237.92823
[30,] 2248.8850 321.80133 276.21372 394.7952 7185.440 8607.617 213.36114
[31,] 1727.0878 841.05727 801.26421 909.5076 6688.014 8079.612 740.42250
[32,] 1732.4413 834.58640 793.38371 895.7423 6687.899 8089.298 729.84879
[33,] 2520.5448 129.47439 157.79092 362.9291 7480.207 8853.898 231.58619
[34,] 2539.0374 80.27733 113.54621 314.9292 7491.759 8879.259 196.59326
[35,] 2333.4441 263.14831 247.34670 438.9599 7295.735 8668.532 250.91092
[36,] 2509.4456 184.52510 212.20980 418.2590 7476.526 8833.750 280.55808
[37,] 2583.6256 87.31627 136.65750 321.1362 7537.310 8921.882 228.31817
[38,] 2490.2285 153.90110 173.41804 381.8631 7452.037 8821.835 236.53603
[,15] [,16] [,17] [,18] [,19] [,20] [,21]
[1,] 6595.123 1865.3315 2942.86395 2284.4748 220.8113 1344.4340 1983.95024
[2,] 4645.312 3647.4779 4878.55606 4150.5466 1870.8531 3262.1270 3920.06552
[3,] 8690.543 701.0442 769.92299 242.1839 2274.4526 886.6440 321.16427
[4,] 7976.421 362.8637 1484.04142 709.8853 1595.6657 326.8433 608.81729
[5,] 8512.272 665.6060 941.35009 410.4231 2071.9687 679.1105 118.30930
[6,] 6896.190 1479.6803 2592.54911 1906.6337 417.0555 979.6784 1633.18616
[7,] 8975.442 1091.0731 508.15393 605.4257 2513.5659 1130.2582 467.91793
[8,] 7227.139 1016.3785 2226.05197 1479.3499 832.9829 627.6409 1284.29706
[9,] 9792.275 1746.3614 344.01278 1173.3057 3351.9820 1960.3680 1303.79415
[10,] 9748.679 1695.1592 309.59297 1122.1255 3313.5827 1921.1816 1267.20906
[11,] 9815.308 1723.1184 461.98339 1157.3096 3409.8612 2018.4362 1380.36649
[12,] 2582.329 5774.7627 7178.90407 6342.7782 4365.9185 5625.2551 6273.27184
[13,] 1734.790 7350.5144 8573.32427 7857.7437 5568.2539 6960.9145 7613.54190
[14,] 9676.443 1610.3930 270.82051 1037.9069 3251.1778 1858.2998 1210.28040
[15,] 0.000 8112.5556 9452.66183 8661.7254 6481.5996 7845.2440 8507.53266
[16,] 8112.556 0.0000 1437.22836 573.0685 1846.5475 689.3066 758.37998
[17,] 9452.662 1437.2284 0.00000 868.9826 3008.1379 1616.9379 959.78246
[18,] 8661.725 573.0685 868.98257 0.0000 2305.0466 955.1063 524.52276
[19,] 6481.600 1846.5475 3008.13793 2305.0466 0.0000 1392.8837 2049.26750
[20,] 7845.244 689.3066 1616.93794 955.1063 1392.8837 0.0000 662.43677
[21,] 8507.533 758.3800 959.78246 524.5228 2049.2675 662.4368 0.00000
[22,] 8238.780 822.5400 1264.51222 796.5463 1763.2587 424.5043 323.51823
[23,] 9071.462 1078.8882 381.51319 531.0304 2632.8882 1240.2027 591.45631
[24,] 9894.294 1846.8302 444.34037 1273.7720 3452.4231 2061.2433 1403.69852
[25,] 8525.596 794.7399 948.52336 550.8869 2063.2315 680.5489 36.44411
[26,] 8087.646 706.9710 1386.69598 806.6407 1622.9040 249.8278 426.92147
[27,] 6680.492 1521.3321 2773.41210 2017.7262 382.8363 1166.9559 1828.08301
[28,] 8142.956 682.3088 1327.32927 749.0738 1681.3521 299.6222 367.91537
[29,] 9495.810 1478.5993 43.19264 909.5023 3050.6774 1659.6932 1002.05501
[30,] 9472.466 1428.5316 91.23807 856.1315 3040.1227 1647.3169 997.22955
[31,] 8953.853 987.6859 498.97624 465.6897 2512.4363 1119.9175 470.96377
[32,] 8958.460 977.4069 494.50184 446.7129 2521.5463 1128.6882 485.14831
[33,] 9747.673 1737.4000 302.02436 1167.0390 3293.5526 1906.1452 1244.85793
[34,] 9765.974 1741.2982 313.93411 1169.3582 3316.7400 1927.3899 1267.47767
[35,] 9560.557 1561.9357 127.72058 995.2683 3107.0595 1719.0932 1058.06810
[36,] 9736.005 1743.7049 306.92480 1175.9020 3276.5363 1892.0687 1229.82368
[37,] 9810.652 1786.7848 359.16087 1214.7406 3360.0946 1971.2842 1310.91144
[38,] 9717.299 1712.6388 275.72952 1143.2327 3261.9870 1875.1162 1213.56486
[,22] [,23] [,24] [,25] [,26] [,27] [,28]
[1,] 1682.8464 2573.9399 3385.7172 1994.34195 1557.98016 529.0860 1618.49587
[2,] 3631.9543 4502.2669 5322.8882 3934.06841 3493.75630 2133.4637 3552.15574
[3,] 631.8197 391.9400 1205.4990 339.01960 689.87283 2017.3535 628.35013
[4,] 535.8477 1103.2899 1921.0517 639.96767 381.47787 1311.8841 377.73988
[5,] 403.3527 562.0189 1384.9475 150.59084 466.20664 1832.2855 404.54680
[6,] 1346.2972 2218.4235 3036.7256 1646.69684 1206.52345 362.5993 1265.32705
[7,] 758.8593 209.4108 946.3215 450.82393 891.08724 2295.8963 833.87024
[8,] 1045.6341 1845.3337 2668.7576 1304.88223 876.88545 547.5698 926.98543
[9,] 1607.1303 721.8134 102.2094 1292.34168 1730.70870 3114.5218 1671.32346
[10,] 1573.9309 681.1536 151.8352 1256.95516 1693.86885 3072.3831 1634.07520
[11,] 1697.0319 789.5145 255.4260 1374.81693 1802.38070 3151.6353 1741.33562
[12,] 6042.2390 6799.7827 7608.9763 6296.64961 5874.79159 4473.1325 5923.87949
[13,] 7315.9615 8200.2504 9017.0653 7625.48872 7186.63418 5840.5395 7246.17165
[14,] 1522.3795 619.5957 242.4857 1202.25820 1635.53267 3003.4562 1575.10060
[15,] 8238.7805 9071.4621 9894.2936 8525.59595 8087.64599 6680.4918 8142.95574
[16,] 822.5400 1078.8882 1846.8302 794.73989 706.97099 1521.3321 682.30875
[17,] 1264.5122 381.5132 444.3404 948.52336 1386.69598 2773.4121 1327.32927
[18,] 796.5463 531.0304 1273.7720 550.88695 806.64069 2017.7262 749.07378
[19,] 1763.2587 2632.8882 3452.4231 2063.23155 1622.90398 382.8363 1681.35214
[20,] 424.5043 1240.2027 2061.2433 680.54886 249.82781 1166.9559 299.62217
[21,] 323.5182 591.4563 1403.6985 36.44411 426.92147 1828.0830 367.91537
[22,] 0.0000 907.7480 1705.0813 323.27583 181.02923 1573.7875 159.52543
[23,] 907.7480 0.0000 823.4601 585.30268 1015.96945 2392.8292 955.50498
[24,] 1705.0813 823.4601 0.0000 1391.66048 1830.60554 3216.2744 1771.42582
[25,] 323.2758 585.3027 1391.6605 0.00000 440.40139 1847.2897 383.08893
[26,] 181.0292 1015.9695 1830.6055 440.40139 0.00000 1413.0156 61.67836
[27,] 1573.7875 2392.8292 3216.2744 1847.28968 1413.01558 0.0000 1466.17145
[28,] 159.5254 955.5050 1771.4258 383.08893 61.67836 1466.1714 0.00000
[29,] 1305.7124 424.7052 401.7469 990.45458 1428.97603 2816.4988 1369.71455
[30,] 1309.0420 407.3758 423.9214 988.94629 1422.87523 2796.4385 1362.59798
[31,] 788.6834 120.8280 942.4101 465.71006 895.14590 2274.4359 834.67823
[32,] 805.1105 113.0169 936.3169 481.83601 907.01789 2279.9593 846.19862
[33,] 1538.2945 680.8206 186.9264 1230.33036 1670.65918 3067.3268 1612.56126
[34,] 1564.9361 695.3965 146.1716 1254.12074 1694.01360 3086.2259 1635.39095
[35,] 1353.7445 496.0049 354.3814 1043.96962 1484.16519 2880.1829 1425.87406
[36,] 1517.8506 677.3840 232.5901 1213.86687 1654.18923 3055.5500 1596.79357
[37,] 1606.9307 740.5065 115.4307 1297.16941 1737.25294 3130.7389 1678.79576
[38,] 1506.0953 652.2891 218.9443 1198.75910 1639.13277 3036.8667 1581.16468
[,29] [,30] [,31] [,32] [,33] [,34]
[1,] 2984.77189 2980.81320 2453.11326 2464.60747 3220.97800 3246.89370
[2,] 4921.16834 4909.17812 4382.04263 4390.50160 5164.39654 5187.54221
[3,] 812.94586 782.29345 288.87123 282.60985 1071.20526 1083.26957
[4,] 1527.18422 1497.69629 990.40227 990.88630 1784.10286 1797.85740
[5,] 984.38175 968.56575 442.53958 449.90610 1235.53539 1253.94044
[6,] 2634.99402 2625.78512 2097.80957 2107.50087 2877.01283 2900.53209
[7,] 548.39734 561.64267 158.80855 187.51138 780.63560 806.20842
[8,] 2269.15647 2248.88502 1727.08784 1732.44132 2520.54478 2539.03740
[9,] 301.88723 321.80133 841.05727 834.58640 129.47439 80.27733
[10,] 269.38298 276.21372 801.26421 793.38371 157.79092 113.54621
[11,] 430.71784 394.79523 909.50765 895.74230 362.92910 314.92919
[12,] 7221.86791 7185.43971 6688.01411 6687.89917 7480.20675 7491.75887
[13,] 8615.55689 8607.61711 8079.61224 8089.29779 8853.89792 8879.25874
[14,] 237.92823 213.36114 740.42250 729.84879 231.58619 196.59326
[15,] 9495.81007 9472.46596 8953.85282 8958.45980 9747.67302 9765.97391
[16,] 1478.59926 1428.53160 987.68588 977.40690 1737.40003 1741.29819
[17,] 43.19264 91.23807 498.97624 494.50184 302.02436 313.93411
[18,] 909.50230 856.13152 465.68973 446.71294 1167.03900 1169.35819
[19,] 3050.67742 3040.12273 2512.43633 2521.54634 3293.55259 3316.73997
[20,] 1659.69317 1647.31691 1119.91751 1128.68820 1906.14522 1927.38991
[21,] 1002.05501 997.22955 470.96377 485.14831 1244.85793 1267.47767
[22,] 1305.71244 1309.04202 788.68339 805.11054 1538.29446 1564.93613
[23,] 424.70519 407.37583 120.82802 113.01685 680.82055 695.39652
[24,] 401.74688 423.92138 942.41010 936.31690 186.92641 146.17163
[25,] 990.45458 988.94629 465.71006 481.83601 1230.33036 1254.12074
[26,] 1428.97603 1422.87523 895.14590 907.01789 1670.65918 1694.01360
[27,] 2816.49876 2796.43854 2274.43586 2279.95930 3067.32676 3086.22590
[28,] 1369.71455 1362.59798 834.67823 846.19862 1612.56126 1635.39095
[29,] 0.00000 94.16533 542.06971 537.69183 259.59048 270.75008
[30,] 94.16533 0.00000 528.09295 518.67928 315.96049 313.54900
[31,] 542.06971 528.09295 0.00000 29.69166 795.38450 812.14395
[32,] 537.69183 518.67928 29.69166 0.00000 793.30579 808.34898
[33,] 259.59048 315.96049 795.38450 793.30579 0.00000 49.24029
[34,] 270.75008 313.54900 812.14395 808.34898 49.24029 0.00000
[35,] 92.83720 180.88533 609.06104 607.90798 187.16584 211.32105
[36,] 266.85315 336.03638 788.73849 788.72520 55.38038 104.27289
[37,] 316.00152 358.75783 856.93118 853.41924 72.79048 45.64154
[38,] 234.03956 296.81742 765.97983 764.49539 33.04463 76.23072
[,35] [,36] [,37] [,38]
[1,] 3035.9374 3200.62376 3289.25775 3188.84929
[2,] 4977.9120 5147.15210 5230.93304 5132.81175
[3,] 887.5381 1069.20899 1128.72267 1043.30632
[4,] 1598.7409 1779.06999 1843.18177 1755.38160
[5,] 1048.5180 1225.66626 1298.47612 1205.40900
[6,] 2690.6411 2859.70214 2943.76831 2845.38378
[7,] 595.1790 763.10441 848.50777 748.77841
[8,] 2333.4441 2509.44561 2583.62559 2490.22852
[9,] 263.1483 184.52510 87.31627 153.90110
[10,] 247.3467 212.20980 136.65750 173.41804
[11,] 438.9599 418.25896 321.13618 381.86313
[12,] 7295.7349 7476.52570 7537.30990 7452.03714
[13,] 8668.5317 8833.74995 8921.88244 8821.83479
[14,] 250.9109 280.55808 228.31817 236.53603
[15,] 9560.5572 9736.00519 9810.65201 9717.29864
[16,] 1561.9357 1743.70494 1786.78483 1712.63880
[17,] 127.7206 306.92480 359.16087 275.72952
[18,] 995.2683 1175.90204 1214.74057 1143.23274
[19,] 3107.0595 3276.53628 3360.09464 3261.98696
[20,] 1719.0932 1892.06874 1971.28417 1875.11621
[21,] 1058.0681 1229.82368 1310.91144 1213.56486
[22,] 1353.7445 1517.85063 1606.93074 1506.09533
[23,] 496.0049 677.38396 740.50646 652.28907
[24,] 354.3814 232.59014 115.43067 218.94433
[25,] 1043.9696 1213.86687 1297.16941 1198.75910
[26,] 1484.1652 1654.18923 1737.25294 1639.13277
[27,] 2880.1829 3055.54997 3130.73889 3036.86665
[28,] 1425.8741 1596.79357 1678.79576 1581.16468
[29,] 92.8372 266.85315 316.00152 234.03956
[30,] 180.8853 336.03638 358.75783 296.81742
[31,] 609.0610 788.73849 856.93118 765.97983
[32,] 607.9080 788.72520 853.41924 764.49539
[33,] 187.1658 55.38038 72.79048 33.04463
[34,] 211.3210 104.27289 45.64154 76.23072
[35,] 0.0000 182.24019 253.35087 156.97535
[36,] 182.2402 0.00000 117.53540 45.10909
[37,] 253.3509 117.53540 0.00000 105.66328
[38,] 156.9753 45.10909 105.66328 0.000004.7 Nilai Stress
> stress = sqrt(sum((D - disparities)^2) / sum(D^2))
> cat("Nilai Stress:", stress, "\n")
Nilai Stress: 0.003585185 Hasil perhitungan menggunakan RStudio menunjukkan nilai “Stress” sebesar 0,003585. Berdasarkan kriteria Kruskal, model MDS sangat akurat dan representasi 2 dimensi hampir identik dengan jarak data asli. Dengan demikian, hasil MDS dapat dipercaya dan digunakan untuk analisis lebih lanjut.
5 KESIMPULAN
Berdasarkan hasil analisis Multidimensional Scaling (MDS) terhadap data jumlah desa yang memiliki fasilitas pendidikan di Indonesia tahun 2024, dapat ditarik beberapa kesimpulan utama sebagai berikut:
Pola Penyebaran Desa Berfasilitas Pendidikan: Pola penyebaran desa yang memiliki fasilitas pendidikan menunjukkan disparitas yang didominasi oleh faktor skala. Provinsi-provinsi di Pulau Jawa (Jawa Tengah, Jawa Timur, dan Jawa Barat) memiliki karakteristik penyebaran dan skala jumlah desa yang sangat tinggi, yang secara fundamental berbeda dari provinsi lain di Indonesia. Hal ini terlihat jelas pada matriks jarak yang menunjukkan jarak besar antara provinsi Jawa dan provinsi lainnya.
Hubungan dan Kemiripan Antarprovinsi (Analisis MDS): Analisis MDS berhasil memetakan hubungan antarprovinsi ke dalam ruang dua dimensi dengan Dimensi 1 dan Dimensi 2 sebagai sumbu laten.
Dimensi 1 merepresentasikan perbedaan utama dalam hal skala absolut dan kepadatan laringan fasilitas pendidikan. Dimensi ini berfungsi sebagai pemisah utama, menempatkan Jawa pada posisi ekstrem (sisi kiri) dan sisanya di sisi berlawanan.
Klasterisasi menunjukkan adanya tiga kelompok struktural yang signifikan. Klaster A (Jawa) dipisahkan berdasarkan skala. Klaster C, yang terdiri dari mayoritas provinsi di luar Jawa (termasuk mayoritas provinsi di Kalimantan, Sulawesi, Maluku, dan Papua), menunjukkan model distribusi fasilitas pendidikan yang sangat homogen dan seragam setelah faktor skala dikendalikan.
- Kualitas pemetaan MDS ke dalam ruang dua dimensi dinyatakan sangat baik dan representatif:
Keragaman Kumulatif: Dua dimensi pertama (Dimensi 1 dan Dimensi 2) mampu menjelaskan 99,75% dari total informasi jarak pada data asli.
Nilai Stress: Nilai Stress yang dihasilkan adalah 0,003585. Berdasarkan kriteria Kruskal, nilai ini menunjukkan bahwa representasi dua dimensi yang dihasilkan adalah sangat akurat dan hampir identik dengan struktur ketidaksamaan pada data mentah.
Secara keseluruhan, model MDS metrik berhasil memberikan visualisasi yang valid dan informatif mengenai pola kemiripan antarprovinsi berdasarkan ketersediaan fasilitas pendidikan desa, menegaskan bahwa perbedaan skala antara Jawa dan luar Jawa adalah faktor pembeda yang paling dominan dalam konteks pemerataan fasilitas pendidikan.
6 DAFTAR PUSTAKA
Badan Pusat Statistik. (2024). Jumlah desa yang memiliki fasilitas pendidikan menurut provinsi dan tingkat pendidikan.
Borg, I., Groenen, P. J. F., & Mair, P. (2018). Applied multidimensional scaling and unfolding. Springer.
Cox, T. F., & Cox, M. A. A. (2001). Multidimensional scaling (2nd ed.). Chapman & Hall/CRC.
Härdle, W. K., & Simar, L. (2015). Applied Multivariate Statistical Analysis. Springer.
Kruskal, J. B., & Wish, M. (1978). Multidimensional scaling. Sage Publications.