1 PENDAHULUAN
1.1 Latar Belakang
Bantuan sosial pangan merupakan salah satu instrumen kebijakan yang ditujukan untuk menjamin ketersediaan pangan dan meringankan beban keluarga kurang mampu. Implementasi program ini seringkali berbeda antar wilayah/provinsi, baik dari segi cakupan penerima (Jumlah Keluarga Penerima Manfaat atau KPM) maupun besaran anggaran yang dialokasikan. Perbedaan tersebut tidak hanya penting untuk evaluasi kebijakan, tetapi juga berguna untuk mengidentifikasi pola kemiripan atau klaster provinsi yang memiliki karakteristik penyaluran bantuan serupa.
Analisis konvensional, seperti perbandingan deskriptif atau korelasi sederhana berguna tetapi terbatas dalam menvisualisasikan struktur kemiripan antar-unit (provinsi) ketika variabel yang dianalisis berdimensi ganda. Multidimensional Scaling (MDS) adalah metode multivariat yang efektif untuk memetakan objek (dalam hal ini provinsi) ke dalam ruang berdimensi rendah sehingga kemiripan (atau dissimilaritas) antar-objek dapat divisualisasikan secara geometris. Dengan MDS, hubungan kompleks antar-provinsi berdasarkan jumlah KPM, jumlah penduduk miskin, dan rata-rata lama sekolah (RLS) dapat ditampilkan pada plot dua atau tiga dimensi yang mudah diinterpretasikan.
1.2 Rumusan Masalah
Bagaimana pola kemiripan antar-provinsi di Indonesia berdasarkan jumlah Keluarga Penerima Manfaat (KPM), penduduk miskin, dan rata-rata lama sekolah (RLS)?
1.3 Tujuan
Mengidentifikasi pola kemiripan antar-provinsi di Indonesia berdasarkan jumlah Keluarga Penerima Manfaat (KPM), penduduk miskin, dan rata-rata lama sekolah.
2 TINJAUAN PUSTAKA
2.1 Multidimensional Scaling (MDS)
Multidimensional Scaling (MDS) merupakan sebuah pendekatan analitis dalam statistik multivariat yang berfungsi untuk mengubah informasi abstrak mengenai kedekatan antar objek menjadi sebuah representasi geometris yang konkret. Informasi kedekatan, yang dapat berupa kemiripan (similarity) atau ketidakmiripan (dissimilarity), diolah dan dipetakan ke dalam ruang berdimensi lebih rendah, umumnya dua atau tiga dimensi. Tujuan dari transformasi ini adalah untuk menghasilkan sebuah “peta persepsi” di mana jarak spasial antar objek secara proporsional mencerminkan tingkat kedekatan mereka dalam data asli. Dengan demikian, MDS memfasilitasi visualisasi dan interpretasi pola hubungan yang kompleks di antara sekumpulan objek secara lebih intuitif (Johnson & Wichern, 2007; Kruskal & Wish, 1978).
MDS diklasifikasikan menjadi dua tipe utama berdasarkan skala pengukuran dari data kedekatan yang digunakan sebagai input. Kedua tipe ini, yaitu MDS Metrik dan MDS Non-Metrik, memiliki tujuan yang serupa namun berbeda dalam cara memperlakukan data dan mengukur keberhasilan representasinya. MDS Metrik digunakan ketika data berupa jarak kuantitatif berskala interval atau rasio, dengan tujuan meminimalkan perbedaan numerik antara jarak dalam data asli dengan jarak pada peta, sehingga representasi spasial secara akurat mencerminkan nilai jarak yang sebenarnya. Kedua, MDS Non-Metrik diterapkan pada data kualitatif berskala ordinal seperti peringkat kemiripan; teknik ini berfokus pada mempertahankan urutan kemiripan (monotonisitas) daripada nilai jarak absolutnya, menjadikannya lebih fleksibel untuk data subjektif. Pemilihan antara kedua pendekatan ini sangat bergantung pada skala pengukuran dan karakteristik data yang dimiliki peneliti (Johnson & Wichern, 2007)
2.2 Metode Analisis Multidimensional Scaling
2.2.1 Matriks Jarak (D)
Langkah pertama dalam analisis MDS yaitu membuat matriks jarak atau matriks D. Jarak yang paling sering digunakan adalah jarak Euclidean yang memiliki rumus :
\[ dist(x, y) = \sqrt{ \sum_{i=1}^{n} (x_i - y_i)^2 } \]
2.2.2 Matriks Jarak Kuadrat
Langkah kedua setelah mendapatkan matriks D, hitung matriks \(D^{2}\) dengan elemen \(d^2_{ij}\) kemudian dilanjutkan dengan menghitung elemen-elemen sebagai berikut.
\[\bar{d}_{i\cdot}^{2} \;=\; \frac{1}{n}\sum_{i=1}^{n} d_{ij}^{2}\]\[\bar{d}_{\cdot j}^{2} \;=\; \frac{1}{n}\sum_{j=1}^{n} d_{ij}^{2}\]\[\bar{d}_{\cdot\cdot}^{2} \;=\; \frac{1}{n^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n} d_{ij}^{2} \]
2.2.3 Matriks B
Langkah ketiga setelah mendapatkan seluruh komponen dan matriks \(D^2\) adalah menentukan matriks B. Elemen matriks B dapat dihitung menghitung rumus :
\[ b_{ij} = -\frac{1}{2} \left( d_{ij}^2 - \bar{d}_{i.}^2 - \bar{d}_{.j}^2 + \bar{d}_{..}^2 \right) \] Keterangan:
\(d_{ij}^2\) = Elemen dari matriks jarak kuadrat
\(\bar{d_{i.}^2}\) = Rata-rata jarak kuadrat pada baris ke-i
\(\bar{d_{.j}^2}\) = Rata-rata jarak kuadrat pada kolom ke-j
\(\bar{d_{..}^2}\) = Rata-rata total jarak kuadrat
2.2.4 Dekomposisi Eigen
Langkah selanjutnya menghitung nilai eigen (λ) dan vektor eigen (v). Nilai eigen dan vektor eigen diperoleh dengan menyelesaikan persamaan karakteristik:
\[ \det(B-\lambda I)=0 \]
Setelah mendapatkan nilai eigen, hitung cumulative variance berdasarkan nilai eigen. Dengan cara mengurutkan nilai eigen dari yang terbesar hingga terkecil, lalu menghitung proporsi varians eigen dengan rumus :
\[ \frac{\lambda_i}{\sum_{j=1}^{p}\lambda_j} \]
P merupakan jumlah variabel. Kemudian akumulasi proporsi varians dihitung untuk menentukan jumlah dimensi yang akan digunakan. Kriteria penentuan jumlah dimensi yang sering digunakan biasanya didasarkan pada kumulatif nilai eigen yang bernilai > 0,8.
2.2.5 Koordinat Objek
Koordinat objek pada MDS dihitung melalui proses dekomposisi eigen terhadap matriks jarak. Saat menggunakan tiga dimensi, tiga eigenvalues terbesar beserta eigenvector yang terkait digunakan untuk membentuk posisi setiap objek. Rumus koordinat objek yaitu:
\[F = \tilde{E} \Lambda^{1/2}\]
Keterangan:
Λ = Matriks diagonal dari akar nilai eigen
\(\tilde{E}\) = Matriks vektor eigen yang bersesuaian
Bentuk \(Λ^{1/2}\) direpresentasikan menjadi \(\Lambda^{1/2} = \begin{bmatrix} \sqrt{\lambda_1} & 0 \\ 0 & \sqrt{\lambda_2} \end{bmatrix}\)
2.2.6 Matriks Disparities
Matriks disparities merupakan gambaran jarak antar objek yang dihitung dari koordinat objek (F). Perhitungan jarak ini umumnya menggunakan jarak Euclidean.
2.2.7 Nilai Standarized Residual Sum of Squares (STRESS)
Nilai STRESS digunakan untuk menilai seberapa baik hasil pemetaan dalam MDS. Perhitungan STRESS menggunakan rumus sebagai berikut:
\[ \text{STRESS}(q) = \sqrt{ \frac{ \sum_i \sum_k \left( a_{ik}^{(q)} - \hat{a}_{ik}^{(q)} \right)^2 }{ \sum_i \sum_k \left( a_{ik}^{(q)} \right)^2 } } \]
Penilaian kualitas nilai STRESS dapat dijelaskan melalui kriteria yang dirangkum pada tabel berikut:
| Nilai Stress | Kriteria |
|---|---|
| > 20% | Buruk |
| 10%-20% | Cukup |
| 5.1%-10% | Baik |
| 2.5%-5% | Sangat baik |
| <2.5% | Sempurna |
2.2.8 Koordinat Akhir (Konfigurasi 2D)
Menampilkan hasil pemetaan MDS dalam bentuk dua dimensi, digunakan dua komponen awal dari koordinat objek (F), yaitu kolom pertama dan kedua. Kedua kolom ini mewakili dua dimensi utama yang memiliki kontribusi terbesar dalam menggambarkan variasi jarak antar objek. Dengan memanfaatkan dua dimensi tersebut, konfigurasi akhir dapat divisualisasikan dalam bentuk plot 2D sehingga hubungan kedekatan atau perbedaan antar wilayah dapat terlihat dengan lebih sederhana dan mudah diinterpretasikan.
2.3 Data
Data yang digunakan pada penelitian ini merupakan data sekunder tahun 2024 yang diperoleh melalui website Badan Pusat Statistik. Variabel yang dianalisis adalah realisasi jumlah keluarga penerima manfaat (KPM) bantuan sosial pangan (X1), jumlah penduduk miskin (X2), dan rata-rata lama sekolah (X3) berdasarkan provinsi di Indonesia.
| Provinsi | X1 | X2 | X3 |
| Aceh | 1932814 | 804.53 | 9.64 |
| Sumatera Utara | 3074198 | 1228.01 | 9.93 |
| Sumatera Barat | 1273639 | 345.73 | 9.44 |
| Riau | 1052412 | 492.25 | 9.43 |
| Jambi | 684629 | 265.42 | 8.9 |
| Sumatera Selatan | 1844008 | 984.24 | 8.57 |
| Bengkulu | 552843 | 281.36 | 9.04 |
| Lampung | 2773343 | 941.23 | 8.36 |
| Kepulauan Bangka Belitung | 168012 | 69.95 | 8.33 |
| Kepulauan Riau | 239773 | 138.3 | 10.5 |
| DKI Jakarta | 812769 | 464.93 | 11.49 |
| Jawa Barat | 15381114 | 3848.67 | 8.87 |
| Jawa Tengah | 12319492 | 3704.33 | 8.02 |
| DI Yogyakarta | 1305356 | 445.55 | 9.92 |
| Jawa Timur | 11806155 | 3982.69 | 8.28 |
| Banten | 2197612 | 791.61 | 9.23 |
| Bali | 598381 | 184.43 | 9.54 |
| Nusa Tenggara Barat | 1998584 | 709.01 | 7.87 |
| Nusa Tenggara Timur | 2355814 | 1127.57 | 8.02 |
| Kalimantan Barat | 1185869 | 336.08 | 7.78 |
| Kalimantan Tengah | 361252 | 145.63 | 8.81 |
| Kalimantan Selatan | 613042 | 183.31 | 8.62 |
| Kalimantan Timur | 330300 | 221.34 | 10.02 |
| Kalimantan Utara | 88967 | 47.83 | 9.35 |
| Sulawesi Utara | 523426 | 186.86 | 9.84 |
| Sulawesi Tengah | 835730 | 379.76 | 9.04 |
| Sulawesi Selatan | 2406145 | 736.48 | 8.86 |
| Sulawesi Tenggara | 730416 | 319.71 | 9.42 |
| Gorontalo | 451196 | 177.99 | 8.29 |
| Sulawesi Barat | 418930 | 162.19 | 8.15 |
| Maluku | 384321 | 297.68 | 10.26 |
| Maluku Utara | 207123 | 83.09 | 9.37 |
| Papua Barat | 219207 | 110.16 | 7.86 |
| Papua Barat Daya | 204273 | 102.27 | 8.39 |
| Papua | 319418 | 152.91 | 9.82 |
| Papua Selatan | 257892 | 92.2 | 8.38 |
| Papua Tengah | 900736 | 308.48 | 6.12 |
| Papua Pegunungan | 788349 | 365.44 | 4.21 |
3 SOURCE CODE
3.1 Library
Library readxl adalah sebuah package dalam pemrograman R yang digunakan untuk membaca file Excel (.xls atau .xlsx) langsung ke dalam R tanpa perlu meng-install Microsoft Excel di perangkat. Library ini dapat membantu mengambil data mentah dari spreadsheet dan mengubahnya menjadi data frame yang siap dianalisis. Keunggulan readxl adalah proses pembacaannya yang cepat, tidak bergantung pada Java, dan stabil untuk berbagai ukuran file, sehingga sangat cocok dipakai dalam alur kerja analisis data.
Library MASS (Modern Applied Statistics with S) adalah sebuah package dalam bahasa pemrograman R yang dikembangkan oleh Venables dan Ripley. Library MASS menyediakan fungsi dan kumpulan data untuk statistik matematika termasuk fungsi untuk regresi linear dan nonlinear, menganalisis multivariat, analisis deret waktu, dan banyak fungsi statistik lainnya. Salah satu fungsi yang ada dalam MASS adalah cmdscale(), yang digunakan untuk melakukan Classical Multidimensional Scaling (MDS), yaitu teknik untuk merepresentasikan data jarak atau dissimilaritas dalam ruang berdimensi rendah (misalnya 2D atau 3D). Di dalam kode untuk kasus ini, cmdscale() digunakan untuk menghitung koordinat objek dalam ruang 2 dimensi (2D) berdasarkan matriks jarak yang dihitung sebelumnya. Fungsi cmdscale() mengembalikan koordinat (fit) yang kemudian digunakan untuk visualisasi dengan plot.
3.2 Impor Data
Data yang telah diunduh dari website bps.go.id kemudian dibaca menggunakan perintah read_excel dari package readxl.
> data <- read_excel("D:/kuliah smt 5/KPM.xlsx")
> data
# A tibble: 38 × 4
Provinsi Realisasi Jumlah Kel…¹ Jumlah Penduduk Misk…² Rata-rata Lama Sekol…³
<chr> <dbl> <dbl> <dbl>
1 Aceh 1932814 805. 9.64
2 Sumater… 3074198 1228. 9.93
3 Sumater… 1273639 346. 9.44
4 Riau 1052412 492. 9.43
5 Jambi 684629 265. 8.9
6 Sumater… 1844008 984. 8.57
7 Bengkulu 552843 281. 9.04
8 Lampung 2773343 941. 8.36
9 Kepulau… 168012 70.0 8.33
10 Kepulau… 239773 138. 10.5
# ℹ 28 more rows
# ℹ abbreviated names:
# ¹`Realisasi Jumlah Keluarga Penerima Manfaat (KPM) Bantuan Sosial Pangan (BANSOS PANGAN)`,
# ²`Jumlah Penduduk Miskin (Ribu Jiwa) Menurut Provinsi`,
# ³`Rata-rata Lama Sekolah (Tahun)`3.3 Code
Code di bawah ini merupakan code yang digunakan untuk memetakan dan mengelompokkan provinsi-provinsi di Indonesia berdasarkan indikator yang ada untuk menggambarkan kemiripan kondisi sosial ekonomi berdasarkan matriks jarak menggunakan metode Multidimensional Scaling (MDS).
- Menghitung jarak
Membentuk matriks B
Menghitung nilai eigen dan vektor eigen dari matriks B
Menghitung cumulative variance
Menentukan koordinat
Membentuk plot
> plot(fit, + xlab = "Dimensi 1", + ylab = "Dimensi 2", + pch = 16, cex = 1) > text(fit[,1], fit[,2], labels = 1:nrow(data), pos = 3, cex=0.5) > abline(h = 0, v = 0, col = "blue", lty = 2)
Menghitung disparities
Menghitung nilai STRESS
4 HASIL DAN PEMBAHASAN
4.1 Matriks Jarak
> Data <- data[,-1]
> Data1 <- scale(Data)
> D <- as.matrix(dist(Data1))
> D
1 2 3 4 5 6 7
1 0.0000000 0.5886688 0.5254656 0.4392496 0.8877452 0.8904089 0.8218942
2 0.5886688 0.0000000 1.1041636 1.0278465 1.4561669 1.1890573 1.4019636
3 0.5254656 1.1041636 0.0000000 0.1609594 0.4785616 0.9706874 0.3922345
4 0.4392496 1.0278465 0.1609594 0.0000000 0.4997712 0.8878280 0.4084612
5 0.8877452 1.4561669 0.4785616 0.4997712 0.0000000 0.8415020 0.1212993
6 0.8904089 1.1890573 0.9706874 0.8878280 0.8415020 0.0000000 0.8868586
7 0.8218942 1.4019636 0.3922345 0.4084612 0.1212993 0.8868586 0.0000000
8 1.0793140 1.3137520 1.1494585 1.1007629 1.0106442 0.3216624 1.0773598
9 1.3948944 1.9400541 0.9983571 1.0240954 0.5259001 1.0583423 0.6261467
10 1.0863818 1.4460545 0.9372489 0.9703882 1.3157108 1.8469420 1.2012140
11 1.5786585 1.6224628 1.6792475 1.6794406 2.1193034 2.4531844 2.0053660
12 4.9926825 4.5152083 5.4141186 5.3678889 5.5763038 4.8701765 5.5968031
13 4.3924168 3.9736360 4.7938895 4.7329722 4.8815307 4.1091901 4.9146035
14 0.4642143 0.9392530 0.4037442 0.4084556 0.8690650 1.2357731 0.7670886
15 4.4296416 3.9846619 4.8560028 4.7856270 4.9608089 4.1801869 4.9864817
16 0.3428553 0.7630698 0.5496445 0.4765958 0.7378245 0.5805210 0.7170901
17 0.7383901 1.3104504 0.2667872 0.3482188 0.5281768 1.1840457 0.4189563
18 1.4449168 1.7845619 1.3463829 1.3179063 1.0235392 0.6354192 1.1262056
19 1.3643853 1.5726977 1.4334341 1.3674321 1.2249146 0.4933629 1.2990898
20 1.6012105 2.0422200 1.3522731 1.3535652 0.9263885 0.9358144 1.0438952
21 1.0504949 1.6232930 0.6111931 0.6455078 0.1693072 0.9667513 0.2383763
22 1.1076441 1.6586296 0.7136522 0.7403044 0.2434407 0.8821900 0.3564457
23 0.8095303 1.2893110 0.5597811 0.5906900 0.9190098 1.4751420 0.8030377
24 0.9594050 1.5427872 0.4613055 0.5310601 0.4604298 1.2448426 0.3701396
25 0.7611177 1.2838171 0.4228924 0.4789572 0.7710742 1.3637851 0.6585250
26 0.7227372 1.2937723 0.3512399 0.3429904 0.1677929 0.7757235 0.1284876
27 0.6534771 1.0205529 0.6965708 0.6553063 0.6884937 0.3800314 0.7203905
28 0.6255032 1.2116920 0.1602113 0.1972331 0.4272331 1.0150731 0.3160966
29 1.3380483 1.8646776 0.9810304 0.9961800 0.5090531 0.9342561 0.6203359
30 1.4430931 1.9602939 1.0950670 1.1093970 0.6243677 0.9856537 0.7357428
31 0.8461720 1.2469330 0.7173653 0.7298646 1.1115411 1.5970030 0.9949701
32 0.9082494 1.4913483 0.4103459 0.4814409 0.4464292 1.2130152 0.3493738
33 1.6840407 2.1887484 1.3436118 1.3568623 0.8717953 1.1529472 0.9811440
34 1.3364877 1.8832254 0.9418842 0.9652710 0.4677817 1.0171062 0.5682617
35 0.8182625 1.3469458 0.4579376 0.5123659 0.7651431 1.3896386 0.6518057
36 1.3423825 1.8880049 0.9470198 0.9727753 0.4743509 1.0200805 0.5766079
37 2.9253188 3.2986567 2.7064117 2.7025473 2.2654703 2.1257271 2.3805030
38 4.4568508 4.7848347 4.2619742 4.2540701 3.8212500 3.6181484 3.9353304
8 9 10 11 12 13 14
1 1.0793140 1.39489436 1.0863818 1.578658 4.992683 4.3924168 0.4642143
2 1.3137520 1.94005407 1.4460545 1.622463 4.515208 3.9736360 0.9392530
3 1.1494585 0.99835714 0.9372489 1.679247 5.414119 4.7938895 0.4037442
4 1.1007629 1.02409541 0.9703882 1.679441 5.367889 4.7329722 0.4084556
5 1.0106442 0.52590006 1.3157108 2.119303 5.576304 4.8815307 0.8690650
6 0.3216624 1.05834226 1.8469420 2.453184 4.870177 4.1091901 1.2357731
7 1.0773598 0.62614668 1.2012140 2.005366 5.596803 4.9146035 0.7670886
8 0.0000000 1.15670177 2.0563116 2.655260 4.695503 3.9296076 1.4295608
9 1.1567018 0.00000000 1.7688330 2.610877 5.834351 5.0802503 1.3886026
10 2.0563116 1.76883296 0.0000000 0.886392 5.908281 5.4019376 0.6433212
11 2.6552598 2.61087750 0.8863920 0.000000 5.824618 5.4502377 1.2867687
12 4.6955025 5.83435116 5.9082812 5.824618 0.000000 1.1337920 5.3903112
13 3.9296076 5.08025029 5.4019376 5.450238 1.133792 0.0000000 4.8281351
14 1.4295608 1.38860255 0.6433212 1.286769 5.390311 4.8281351 0.0000000
15 4.0257843 5.18185558 5.4261159 5.430517 1.148726 0.3812973 4.8673557
16 0.7433158 1.18734117 1.3502700 1.912275 4.910062 4.2596763 0.7097791
17 1.3783751 0.99999229 0.7901024 1.614277 5.674484 5.0584337 0.4546483
18 0.5139647 0.91359392 2.2753543 2.978389 5.064320 4.2454207 1.7024221
19 0.3555983 1.26392486 2.3336692 2.937917 4.710488 3.8781041 1.7199359
20 0.8972801 0.59930617 2.2410383 3.026346 5.491986 4.6808995 1.7467598
21 1.1237071 0.40219664 1.3769020 2.210139 5.725890 5.0217159 0.9914943
22 1.0087210 0.29221617 1.5356546 2.355321 5.649379 4.9267828 1.1094325
23 1.6889872 1.38563575 0.4006294 1.230043 5.759912 5.1941460 0.3703761
24 1.4359867 0.83135976 0.9420525 1.804969 5.862664 5.2195154 0.7068604
25 1.5663095 1.23972060 0.5459659 1.375234 5.717884 5.1364136 0.3510446
26 0.9697759 0.68479364 1.2258768 1.997272 5.470558 4.7885264 0.7324944
27 0.4686011 1.02754051 1.5936149 2.208076 4.889823 4.1976288 0.9656840
28 1.2190672 0.93690202 0.9095133 1.692417 5.549020 4.9144229 0.4577777
29 1.0221971 0.14000000 1.8014339 2.624300 5.704322 4.9437128 1.3769691
30 1.0527146 0.18810267 1.9148285 2.739668 5.732236 4.9578720 1.4917486
31 1.8144305 1.58975533 0.2562908 1.023364 5.734981 5.1974266 0.4121656
32 1.4050196 0.84721474 0.9220611 1.777557 5.814819 5.1747553 0.6593477
33 1.1881698 0.38521305 2.1503628 2.982911 5.838366 5.0362498 1.7400260
34 1.1252832 0.05963291 1.7189187 2.557132 5.801762 5.0522703 1.3318071
35 1.5961646 1.21719929 0.5545056 1.403203 5.782123 5.1946503 0.4180737
36 1.1226357 0.05325768 1.7272812 2.565617 5.797181 5.0483252 1.3384805
37 2.0068837 1.82822784 3.5765522 4.376544 5.939049 4.9994955 3.1002238
38 3.4770952 3.37349674 5.1304926 5.930101 6.673017 5.6618855 4.6536578
15 16 17 18 19 20 21
1 4.4296416 0.3428553 0.7383901 1.4449168 1.3643853 1.6012105 1.0504949
2 3.9846619 0.7630698 1.3104504 1.7845619 1.5726977 2.0422200 1.6232930
3 4.8560028 0.5496445 0.2667872 1.3463829 1.4334341 1.3522731 0.6111931
4 4.7856270 0.4765958 0.3482188 1.3179063 1.3674321 1.3535652 0.6455078
5 4.9608089 0.7378245 0.5281768 1.0235392 1.2249146 0.9263885 0.1693072
6 4.1801869 0.5805210 1.1840457 0.6354192 0.4933629 0.9358144 0.9667513
7 4.9864817 0.7170901 0.4189563 1.1262056 1.2990898 1.0438952 0.2383763
8 4.0257843 0.7433158 1.3783751 0.5139647 0.3555983 0.8972801 1.1237071
9 5.1818556 1.1873412 0.9999923 0.9135939 1.2639249 0.5993062 0.4021966
10 5.4261159 1.3502700 0.7901024 2.2753543 2.3336692 2.2410383 1.3769020
11 5.4305171 1.9122754 1.6142769 2.9783893 2.9379168 3.0263457 2.2101385
12 1.1487260 4.9100624 5.6744843 5.0643202 4.7104875 5.4919860 5.7258903
13 0.3812973 4.2596763 5.0584337 4.2454207 3.8781041 4.6808995 5.0217159
14 4.8673557 0.7097791 0.4546483 1.7024221 1.7199359 1.7467598 0.9914943
15 0.0000000 4.3179534 5.1171792 4.3614273 3.9725463 4.8042374 5.1057508
16 4.3179534 0.0000000 0.8070571 1.1122696 1.0428830 1.3002195 0.9066907
17 5.1171792 0.8070571 0.0000000 1.5142348 1.6407786 1.4515392 0.5997741
18 4.3614273 1.1122696 1.5142348 0.0000000 0.4506042 0.4489429 1.0640140
19 3.9725463 1.0428830 1.6407786 0.4506042 0.0000000 0.8873070 1.3129092
20 4.8042374 1.3002195 1.4515392 0.4489429 0.8873070 0.0000000 0.8929887
21 5.1057508 0.9066907 0.5997741 1.0640140 1.3129092 0.8929887 0.0000000
22 5.0189834 0.9120814 0.7493139 0.9021168 1.1816160 0.7205375 0.1751991
23 5.2313946 1.0175506 0.4002913 1.8815607 1.9567184 1.8447292 0.9884752
24 5.2858173 0.9706970 0.2541265 1.4836323 1.6686877 1.3490729 0.4575228
25 5.1835528 0.9230341 0.2453106 1.7414797 1.8370421 1.6953648 0.8412272
26 4.8590639 0.5925232 0.4573784 1.0635317 1.2041767 1.0321441 0.3308541
27 4.2774537 0.3123010 0.9429247 0.8153636 0.7894116 1.0299577 0.8399213
28 4.9725039 0.6555808 0.1718916 1.3718139 1.4772974 1.3422998 0.5375268
29 5.0458115 1.1058618 1.0189834 0.7766282 1.1249087 0.4930064 0.4255650
30 5.0650604 1.1999754 1.1335077 0.7508944 1.1266309 0.4131612 0.5380593
31 5.2230641 1.1073501 0.6005859 2.0441840 2.0860127 2.0334996 1.1908543
32 5.2400378 0.9237273 0.2058971 1.4690503 1.6433253 1.3498337 0.4625741
33 5.1527029 1.4293955 1.3747156 0.7927581 1.2032018 0.3663397 0.7756479
34 5.1509532 1.1323082 0.9471614 0.9065706 1.2422605 0.6187885 0.3478240
35 5.2420763 0.9692923 0.2439884 1.7526322 1.8597482 1.6903910 0.8227239
36 5.1484181 1.1357076 0.9544199 0.9009109 1.2410882 0.6092360 0.3555810
37 5.1699584 2.6063714 2.7896139 1.5148955 1.8031858 1.3548004 2.2025530
38 5.8657918 4.1311925 4.3452272 3.0212714 3.2284950 2.9100388 3.7550648
22 23 24 25 26 27 28
1 1.1076441 0.8095303 0.95940496 0.76111770 0.7227372 0.6534771 0.6255032
2 1.6586296 1.2893110 1.54278723 1.28381713 1.2937723 1.0205529 1.2116920
3 0.7136522 0.5597811 0.46130549 0.42289243 0.3512399 0.6965708 0.1602113
4 0.7403044 0.5906900 0.53106007 0.47895716 0.3429904 0.6553063 0.1972331
5 0.2434407 0.9190098 0.46042983 0.77107417 0.1677929 0.6884937 0.4272331
6 0.8821900 1.4751420 1.24484259 1.36378510 0.7757235 0.3800314 1.0150731
7 0.3564457 0.8030377 0.37013960 0.65852498 0.1284876 0.7203905 0.3160966
8 1.0087210 1.6889872 1.43598670 1.56630946 0.9697759 0.4686011 1.2190672
9 0.2922162 1.3856357 0.83135976 1.23972060 0.6847936 1.0275405 0.9369020
10 1.5356546 0.4006294 0.94205253 0.54596594 1.2258768 1.5936149 0.9095133
11 2.3553214 1.2300427 1.80496880 1.37523423 1.9972722 2.2080756 1.6924175
12 5.6493790 5.7599122 5.86266373 5.71788369 5.4705581 4.8898227 5.5490202
13 4.9267828 5.1941460 5.21951545 5.13641364 4.7885264 4.1976288 4.9144229
14 1.1094325 0.3703761 0.70686036 0.35104458 0.7324944 0.9656840 0.4577777
15 5.0189834 5.2313946 5.28581727 5.18355281 4.8590639 4.2774537 4.9725039
16 0.9120814 1.0175506 0.97069700 0.92303405 0.5925232 0.3123010 0.6555808
17 0.7493139 0.4002913 0.25412647 0.24531055 0.4573784 0.9429247 0.1718916
18 0.9021168 1.8815607 1.48363234 1.74147972 1.0635317 0.8153636 1.3718139
19 1.1816160 1.9567184 1.66868769 1.83704208 1.2041767 0.7894116 1.4772974
20 0.7205375 1.8447292 1.34907289 1.69536480 1.0321441 1.0299577 1.3422998
21 0.1751991 0.9884752 0.45752283 0.84122719 0.3308541 0.8399213 0.5375268
22 0.0000000 1.1438136 0.62856079 0.99398313 0.4003155 0.7857597 0.6667345
23 1.1438136 0.0000000 0.57718488 0.16068707 0.8269749 1.2338073 0.5118714
24 0.6285608 0.5771849 0.00000000 0.44119655 0.4712119 1.0436926 0.3355243
25 0.9939831 0.1606871 0.44119655 0.00000000 0.6858523 1.1135121 0.3721244
26 0.4003155 0.8269749 0.47121190 0.68585232 0.0000000 0.5974397 0.3168083
27 0.7857597 1.2338073 1.04369259 1.11351208 0.5974397 0.0000000 0.7868554
28 0.6667345 0.5118714 0.33552433 0.37212443 0.3168083 0.7868554 0.0000000
29 0.2728746 1.4101210 0.87947610 1.26261390 0.6532578 0.9227827 0.9348217
30 0.3874788 1.5244144 0.98872438 1.37698938 0.7667625 0.9995895 1.0503138
31 1.3422935 0.2106065 0.78730035 0.36203878 1.0055883 1.3555598 0.6917932
32 0.6301609 0.5485341 0.05190216 0.40720724 0.4409902 1.0049582 0.2851294
33 0.6336850 1.7630761 1.21574635 1.61687126 1.0144386 1.2088109 1.2964059
34 0.2360882 1.3334618 0.78450648 1.18762507 0.6258146 0.9797841 0.8802326
35 0.9815134 0.1768786 0.40269488 0.07013316 0.6913923 1.1493315 0.3853061
36 0.2391238 1.3421817 0.79279360 1.19539063 0.6329813 0.9796840 0.8880278
37 2.0417352 3.1818807 2.65416149 3.03421905 2.3793722 2.3141832 2.6881858
38 3.5967895 4.7360759 4.20337281 4.58955727 3.9338805 3.8342853 4.2436068
29 30 31 32 33 34 35
1 1.3380483 1.4430931 0.8461720 0.90824943 1.6840407 1.33648767 0.81826254
2 1.8646776 1.9602939 1.2469330 1.49134832 2.1887484 1.88322541 1.34694584
3 0.9810304 1.0950670 0.7173653 0.41034587 1.3436118 0.94188423 0.45793765
4 0.9961800 1.1093970 0.7298646 0.48144093 1.3568623 0.96527101 0.51236591
5 0.5090531 0.6243677 1.1115411 0.44642920 0.8717953 0.46778170 0.76514306
6 0.9342561 0.9856537 1.5970030 1.21301524 1.1529472 1.01710618 1.38963859
7 0.6203359 0.7357428 0.9949701 0.34937381 0.9811440 0.56826172 0.65180575
8 1.0221971 1.0527146 1.8144305 1.40501956 1.1881698 1.12528323 1.59616457
9 0.1400000 0.1881027 1.5897553 0.84721474 0.3852130 0.05963291 1.21719929
10 1.8014339 1.9148285 0.2562908 0.92206113 2.1503628 1.71891873 0.55450560
11 2.6242999 2.7396684 1.0233642 1.77755680 2.9829113 2.55713204 1.40320306
12 5.7043218 5.7322359 5.7349813 5.81481855 5.8383661 5.80176167 5.78212340
13 4.9437128 4.9578720 5.1974266 5.17475530 5.0362498 5.05227034 5.19465027
14 1.3769691 1.4917486 0.4121656 0.65934770 1.7400260 1.33180707 0.41807369
15 5.0458115 5.0650604 5.2230641 5.24003782 5.1527029 5.15095319 5.24207626
16 1.1058618 1.1999754 1.1073501 0.92372730 1.4293955 1.13230816 0.96929228
17 1.0189834 1.1335077 0.6005859 0.20589715 1.3747156 0.94716144 0.24398837
18 0.7766282 0.7508944 2.0441840 1.46905035 0.7927581 0.90657062 1.75263217
19 1.1249087 1.1266309 2.0860127 1.64332526 1.2032018 1.24226049 1.85974815
20 0.4930064 0.4131612 2.0334996 1.34983368 0.3663397 0.61878854 1.69039099
21 0.4255650 0.5380593 1.1908543 0.46257407 0.7756479 0.34782398 0.82272389
22 0.2728746 0.3874788 1.3422935 0.63016093 0.6336850 0.23608818 0.98151341
23 1.4101210 1.5244144 0.2106065 0.54853412 1.7630761 1.33346177 0.17687860
24 0.8794761 0.9887244 0.7873004 0.05190216 1.2157463 0.78450648 0.40269488
25 1.2626139 1.3769894 0.3620388 0.40720724 1.6168713 1.18762507 0.07013316
26 0.6532578 0.7667625 1.0055883 0.44099015 1.0144386 0.62581456 0.69139235
27 0.9227827 0.9995895 1.3555598 1.00495821 1.2088109 0.97978415 1.14933149
28 0.9348217 1.0503138 0.6917932 0.28512940 1.2964059 0.88023264 0.38530606
29 0.0000000 0.1155055 1.6091128 0.88759894 0.3630693 0.13246527 1.24695908
30 0.1155055 0.0000000 1.7239355 0.99870485 0.2487364 0.21378528 1.36048112
31 1.6091128 1.7239355 0.0000000 0.75808941 1.9643680 1.53656299 0.38730056
32 0.8875989 0.9987048 0.7580894 0.00000000 1.2301376 0.79840223 0.37459299
33 0.3630693 0.2487364 1.9643680 1.23013764 0.0000000 0.43175718 1.59717987
34 0.1324653 0.2137853 1.5365630 0.79840223 0.4317572 0.00000000 1.16626501
35 1.2469591 1.3604811 0.3873006 0.37459299 1.5971799 1.16626501 0.00000000
36 0.1263201 0.2054850 1.5455139 0.80649900 0.4240513 0.02024455 1.17454422
37 1.7770141 1.6657372 3.3751827 2.66426112 1.4446694 1.87131633 3.02224367
38 3.3297673 3.2172523 4.9293274 4.21555367 2.9883933 3.41890035 4.57612329
36 37 38
1 1.34238250 2.925319 4.456851
2 1.88800490 3.298657 4.784835
3 0.94701984 2.706412 4.261974
4 0.97277533 2.702547 4.254070
5 0.47435094 2.265470 3.821250
6 1.02008049 2.125727 3.618148
7 0.57660791 2.380503 3.935330
8 1.12263566 2.006884 3.477095
9 0.05325768 1.828228 3.373497
10 1.72728124 3.576552 5.130493
11 2.56561745 4.376544 5.930101
12 5.79718134 5.939049 6.673017
13 5.04832524 4.999495 5.661886
14 1.33848048 3.100224 4.653658
15 5.14841810 5.169958 5.865792
16 1.13570765 2.606371 4.131193
17 0.95441991 2.789614 4.345227
18 0.90091087 1.514895 3.021271
19 1.24108823 1.803186 3.228495
20 0.60923604 1.354800 2.910039
21 0.35558099 2.202553 3.755065
22 0.23912381 2.041735 3.596790
23 1.34218171 3.181881 4.736076
24 0.79279360 2.654161 4.203373
25 1.19539063 3.034219 4.589557
26 0.63298128 2.379372 3.933881
27 0.97968401 2.314183 3.834285
28 0.88802783 2.688186 4.243607
29 0.12632015 1.777014 3.329767
30 0.20548495 1.665737 3.217252
31 1.54551391 3.375183 4.929327
32 0.80649900 2.664261 4.215554
33 0.42405128 1.444669 2.988393
34 0.02024455 1.871316 3.418900
35 1.17454422 3.022244 4.576123
36 0.00000000 1.862812 3.410859
37 1.86281221 0.000000 1.557010
38 3.41085852 1.557010 0.000000Matriks dissimilaritas (D) menunjukkan jarak antar observasi yang merepresentasikan perbedaan profil antar provinsi; nilai diagonal bernilai nol dan matriks bersifat simetris. Nilai jarak dalam matriks ini berkisar dari nilai kecil hingga relatif besar (menunjukkan variasi yang nyata antar provinsi), di mana nilai yang lebih kecil menunjukkan kemiripan karakteristik antar pasangan provinsi, sedangkan nilai yang lebih besar menunjukkan perbedaan yang signifikan. Analisis ringkas terhadap distribusi jarak memberikan gambaran tentang sebaran kemiripan. Sejumlah pasangan provinsi memiliki jarak sangat kecil (serupa), sementara pasangan lain menunjukkan jarak besar (kontras).
4.2 Nilai Eigen dan Vektor Eigen
> eigen_result <- eigen(B)
> eigenvalues <- eigen_result$values
> eigenvalues
[1] 7.419107e+01 3.621951e+01 5.894216e-01 7.290920e-15 7.025028e-15
[6] 4.615614e-15 3.396180e-15 3.297190e-15 3.183131e-15 2.733317e-15
[11] 2.638158e-15 2.157089e-15 2.137223e-15 2.087198e-15 1.478019e-15
[16] 1.268202e-15 9.448780e-16 7.220292e-16 6.116622e-16 5.696039e-16
[21] 2.249469e-16 9.332880e-17 -3.172459e-16 -4.642466e-16 -5.030892e-16
[26] -5.329788e-16 -7.373780e-16 -7.831710e-16 -8.224447e-16 -8.774243e-16
[31] -1.538432e-15 -1.651970e-15 -1.991772e-15 -2.283944e-15 -3.326787e-15
[36] -3.466385e-15 -5.442495e-15 -9.691608e-15
> eigenvectors <- eigen_result$vectors
> eigenvectors
[,1] [,2] [,3] [,4] [,5]
[1,] -0.0005946813 -0.109369431 0.130050547 0.7220593318 0.000000000
[2,] -0.0581245856 -0.160974798 0.217667314 -0.0219057121 0.337789734
[3,] 0.0497030426 -0.071532417 -0.119027695 0.0451193004 -0.039462173
[4,] 0.0427916454 -0.071644154 0.075758246 -0.0626077673 -0.377023771
[5,] 0.0627988769 0.005005022 -0.035378976 0.0466783883 0.299510174
[6,] -0.0277045075 0.031280004 0.322357942 0.0527283338 -0.191749295
[7,] 0.0664929468 -0.013384930 0.014253993 0.0845952410 -0.110672670
[8,] -0.0488900948 0.055671393 0.035153873 -0.0271154001 -0.189063916
[9,] 0.0832001113 0.087189371 -0.077697116 -0.0895781308 -0.079037963
[10,] 0.1053447053 -0.204963081 -0.037710351 -0.0204364777 -0.312409018
[11,] 0.0785892286 -0.345922519 0.110313679 -0.0416041968 -0.079091075
[12,] -0.5761544648 -0.123238539 -0.632462408 0.2086048908 0.021256333
[13,] -0.5038798660 0.007818205 0.051627516 -0.4219440600 -0.045975773
[14,] 0.0473043218 -0.137686627 -0.035908703 0.1060799788 0.031887815
[15,] -0.5110395099 -0.029313132 0.445945997 0.2627869010 -0.036259415
[16,] -0.0113241515 -0.055509762 0.048282377 0.0938371077 -0.102250082
[17,] 0.0801250491 -0.078905992 -0.088690958 0.0787985589 0.025566241
[18,] -0.0183292914 0.129014616 0.027439943 0.0954688899 0.047874249
[19,] -0.0589145240 0.100018447 0.319818716 0.0151139835 0.055743105
[20,] 0.0300504397 0.151383615 -0.101338994 0.1190648122 0.050335511
[21,] 0.0789764363 0.020672242 -0.060010859 0.0798310785 0.228964093
[22,] 0.0673988830 0.044260543 -0.091845542 -0.0294699111 0.030988897
[23,] 0.0899195241 -0.142537056 0.016049114 -0.0334937516 0.374088477
[24,] 0.1006960306 -0.048661482 -0.079111008 -0.0044139269 0.001696635
[25,] 0.0857563342 -0.118766695 -0.067040853 -0.0098728810 -0.023809506
[26,] 0.0517960616 -0.016436069 0.030032030 -0.0059417888 0.102089491
[27,] -0.0167374179 -0.006011814 -0.057682540 0.1354254812 -0.132131745
[28,] 0.0643337477 -0.065792429 0.001715688 0.0215136795 0.064151912
[29,] 0.0671658508 0.089328578 -0.052986362 0.0879735365 -0.138441562
[30,] 0.0673182546 0.108505717 -0.058754672 0.0643895029 -0.016423218
[31,] 0.0856620140 -0.176251659 0.071929121 -0.0368376847 0.034046440
[32,] 0.0953060002 -0.052514258 -0.077971750 0.0370795438 -0.122682799
[33,] 0.0723368895 0.149200796 -0.053164758 0.1559657182 -0.019372250
[34,] 0.0805186128 0.078423943 -0.057527914 0.0932504048 0.220175554
[35,] 0.0930582030 -0.114523202 -0.044078617 0.0003332368 -0.079176231
[36,] 0.0799453511 0.079678105 -0.081135314 0.0383385765 -0.303721094
[37,] 0.0165065591 0.375557777 -0.047670321 0.1743794193 -0.134631903
[38,] -0.0114020254 0.630931673 0.038799613 -0.0440468354 0.081562981
[,6] [,7] [,8] [,9] [,10]
[1,] 0.000000000 0.00000000 0.000000000 0.000000000 0.0000000000
[2,] -0.146640072 -0.25746604 0.387439262 0.113154762 0.0178322938
[3,] 0.186331467 -0.11684210 -0.064200957 -0.008644779 -0.3916763937
[4,] 0.085849821 0.03754163 0.024443241 0.273859317 -0.1984321923
[5,] -0.475222936 0.24218240 -0.216694384 -0.123456370 -0.0528214713
[6,] -0.090102702 0.30532243 -0.120399994 -0.040027901 -0.1132107361
[7,] -0.034716397 0.03796556 -0.053115972 -0.082983891 -0.5423417590
[8,] -0.165365467 0.19455146 -0.297272812 0.208374602 0.2243688368
[9,] 0.046507665 0.15975856 0.061355367 0.190200820 -0.0924459559
[10,] -0.343196377 -0.11375988 0.089987478 -0.343889320 0.0225677946
[11,] -0.027744636 -0.23222502 -0.287357174 0.328573035 0.1407825245
[12,] -0.055118067 -0.02947220 -0.088758534 -0.007094809 0.1181991682
[13,] -0.074236073 0.06809526 0.078093548 0.026042811 -0.2291467423
[14,] -0.173209692 -0.06449752 0.029481991 0.382550429 -0.1531049203
[15,] 0.168778757 0.13855912 0.076822895 -0.047405928 -0.0464625004
[16,] -0.178464936 -0.06167533 0.265253614 -0.118463068 -0.0074300260
[17,] 0.162360361 0.17752557 0.152275119 -0.083858168 -0.1429090573
[18,] -0.148533811 -0.04122786 -0.304070083 -0.011000947 -0.0745646287
[19,] -0.163779327 -0.24097490 -0.109550214 -0.091537977 0.1206387346
[20,] -0.064698202 0.25371613 0.009679062 0.141595601 -0.1116844353
[21,] 0.191665789 0.11366938 -0.015641758 0.223016558 -0.1494444234
[22,] 0.042001308 0.02586254 0.138933582 -0.050555608 0.0001740547
[23,] -0.019865470 0.12083606 -0.035324679 -0.140737554 -0.0955353380
[24,] -0.287942638 0.06395252 0.267206022 0.184745577 -0.0279771354
[25,] 0.118805223 0.24081308 0.184037950 -0.101249589 0.1720865235
[26,] -0.086070095 0.01816587 -0.010664705 0.107473556 0.0237422151
[27,] -0.186837084 0.09395019 -0.029679849 -0.047128559 0.0128263769
[28,] 0.004246845 0.03727605 -0.240820794 -0.052704277 -0.0286971363
[29,] 0.171503741 -0.14098129 -0.113837148 -0.035446286 0.0452913337
[30,] -0.094671649 -0.02020729 -0.034116227 -0.153113288 -0.1167209096
[31,] 0.240861707 0.36947400 -0.003127081 -0.037190896 0.3991084463
[32,] -0.059192069 -0.17849814 0.018831885 -0.118019017 0.0509111622
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[25,] 0.2034687871 0.004337747 -0.062156090
[26,] 0.1628488497 -0.002060962 0.053240626
[27,] -0.0145679863 -0.135618719 -0.114352315
[28,] -0.0239318955 -0.034168517 0.032025919
[29,] -0.0579261654 -0.069816293 -0.024427748
[30,] 0.1004448648 -0.040177753 0.112374188
[31,] -0.1524350567 -0.002953948 -0.062294628
[32,] 0.0750347726 -0.033281956 -0.045217980
[33,] 0.0787899564 -0.133218503 -0.093197358
[34,] 0.1098742742 -0.076383583 0.077964180
[35,] -0.0605740755 -0.010405267 -0.097112500
[36,] 0.1622984254 -0.012479379 -0.019238254
[37,] 0.1237277189 -0.117244168 -0.016902010
[38,] 0.0111026737 0.142784027 -0.029874653Dari hasil eigen-dekomposisi matriks B diperoleh nilai eigen terbesar sebesar 74.19107 dan 36.21951 serta eigenvalue ketiga sebesar 0.58942. Jumlah ketiga eigenvalue tersebut menyumbang seluruh variasi informasi yang bermakna karena sisanya memiliki nilai yang sangat kecil atau numerik nol. Kontribusi kumulatif dua dimensi pertama mencapai sekitar 99.47 persen sehingga embedding dua dimensi cukup untuk merepresentasikan struktur kemiripan antar provinsi. Nilai-nilai eigenvalue yang sangat kecil dan beberapa angka negatif yang muncul adalah akibat ketidakpastian numerik pada perhitungan floating point dan oleh karena itu diperlakukan sebagai nol untuk tujuan konstruksi koordinat MDS dan interpretasi.
4.3 Cumulative Variance
> cumulative_variance <- cumsum(eigenvalues) / sum(eigenvalues)
> cumulative_variance
[1] 0.6683880 0.9946899 1.0000000 1.0000000 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 1.0000000Output MDS memperlihatkan proporsi kumulatif varians yang dijelaskan oleh komponen utama dari Gram matrix. Nilai menunjukkan bahwa Dimensi 1 menyumbang sekitar 66,84 persen dari variasi, dan bahwa Dimensi 1 ditambah Dimensi 2 menyumbang sekitar 99,47 persen. Karena data hanya memiliki tiga variabel, tiga eigenvalue pertama sudah mencakup semua variasi yang bermakna, sehingga penggunaan dua dimensi dalam MDS sudah sangat memadai untuk merepresentasikan hampir seluruh informasi dalam data.
4.4 Plot
> plot(fit,
+ xlab = "Dimensi 1",
+ ylab = "Dimensi 2",
+ pch = 16, cex = 1)
> text(fit[,1], fit[,2], labels = 1:nrow(data), pos = 3, cex=0.5)
> abline(h = 0, v = 0, col = "blue", lty = 2)
Hasil MDS dua dimensi menunjukkan fit yang sangat baik (stress ≈ 0,0088), sehingga posisi relatif titik pada plot dapat diandalkan untuk menilai kemiripan antar-provinsi. Secara umum tampak tiga pola utama. Pertama, terdapat satu gugus provinsi yang terletak jauh di sebelah kiri pada Dimensi 1, yaitu Jawa Barat, Jawa Timur, dan Jawa Tengah. Ketiga provinsi ini berdekatan satu sama lain, sehingga dapat diinterpretasikan memiliki profil yang serupa menurut variabel yang dianalisis. Kedua, di sekitar pusat (nilai Dimensi 1 mendekati nol) terdapat kelompok provinsi lain dari Sumatra dan Nusa Tenggara (mis. Sumatera Utara, Lampung, Sumatera Selatan, Nusa Tenggara Barat, Nusa Tenggara Timur) yang menunjukkan kemiripan moderat satu sama lain tetapi berbeda dari kelompok Jawa. Ketiga, di sisi kanan-bawah terdapat kumpulan provinsi yang relatif serupa di antara mereka sendiri, termasuk Kepulauan Bangka Belitung, Kepulauan Riau, Kalimantan Timur, Kalimantan Utara, Maluku, Maluku Utara, Papua, Papua Barat Daya, Sulawesi Utara yang menempati area ber-Dimensi1 positif dan pada umumnya Dimensi2 negatif atau kecil; posisi ini mengindikasikan karakteristik kelompok yang berbeda dari kedua gugus sebelumnya.
Selain pola kelompok tersebut, terdapat outlier yang jelas pada arah Dimensi 2, yaitu Papua Pegunungan dan Papua Tengah menonjol jauh ke atas (Dimensi 2 sangat positif), yang berarti kedua provinsi ini memiliki komposisi karakteristik yang berbeda secara substansial dibandingkan mayoritas provinsi lain (mereka unik pada variabel-variabel yang digunakan).
4.5 Disparities
> 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,] 0.0000000 0.5848129 0.48943476 0.43726739 0.8786131 0.8780831 0.8170720
[2,] 0.5848129 0.0000000 1.07347959 1.02205606 1.4431493 1.1863377 1.3932385
[3,] 0.4894348 1.0734796 0.00000000 0.05953459 0.4742330 0.9096164 0.3786521
[4,] 0.4372674 1.0220561 0.05953459 0.00000000 0.4924337 0.8674071 0.4057227
[5,] 0.8786131 1.4431493 0.47423302 0.49243373 0.0000000 0.7954206 0.1151586
[6,] 0.8780831 1.1863377 0.90961644 0.86740706 0.7954206 0.0000000 0.8547313
[7,] 0.8170720 1.3932385 0.37865208 0.40572273 0.1151586 0.8547313 0.0000000
[8,] 1.0768523 1.3062580 1.14334735 1.10032137 1.0091925 0.2341955 1.0772403
[9,] 1.3857458 1.9267560 0.99785275 1.01729611 0.5248955 1.0127956 0.6221544
[10,] 1.0787200 1.4327013 0.93516732 0.96647005 1.3157096 1.8261371 1.2005513
[11,] 1.5785857 1.6203681 1.66999100 1.67923106 2.1163496 2.4477769 2.0040094
[12,] 4.9582430 4.4677868 5.39974999 5.34028009 5.5574301 4.8146915 5.5747362
[13,] 4.3920042 3.9715908 4.79209874 4.73293591 4.8810736 4.1039300 4.9145198
[14,] 0.4463863 0.9188558 0.39866928 0.39935716 0.8690649 1.2047738 0.7661213
[15,] 4.4229974 3.9808059 4.83659207 4.77718032 4.9470266 4.1791099 4.9754554
[16,] 0.3370591 0.7519071 0.53442459 0.47612878 0.7350235 0.5410441 0.7166140
[17,] 0.7190392 1.2891703 0.26576863 0.32452468 0.5265886 1.1412164 0.4114340
[18,] 1.4427677 1.7785758 1.34167889 1.31738408 1.0224024 0.5937101 1.1261601
[19,] 1.3565843 1.5707411 1.39327628 1.35453364 1.1941739 0.4933590 1.2777324
[20,] 1.5913255 2.0274812 1.35220490 1.34671916 0.9250033 0.8774601 1.0401161
[21,] 1.0403113 1.6092336 0.60951129 0.63703641 0.1682477 0.9211032 0.2314572
[22,] 1.0944649 1.6415196 0.71334696 0.72903572 0.2395496 0.8228824 0.3470134
[23,] 0.8047850 1.2799855 0.55009132 0.58890851 0.9181612 1.4562765 0.8030365
[24,] 0.9458708 1.5258695 0.46028644 0.51757881 0.4592041 1.2060811 0.3631326
[25,] 0.7459250 1.2650726 0.42100477 0.46624106 0.7706909 1.3306145 0.6555606
[26,] 0.7186464 1.2857274 0.33207412 0.34118909 0.1601017 0.7425485 0.1279154
[27,] 0.6373845 0.9984186 0.69497679 0.64724871 0.6882807 0.2435020 0.7182704
[28,] 0.6176946 1.2002957 0.13066943 0.18886377 0.4262828 0.9847711 0.3159500
[29,] 1.3306488 1.8530637 0.97971929 0.99126429 0.5088735 0.8887040 0.6181842
[30,] 1.4357947 1.9487727 1.09408887 1.10458000 0.6241097 0.9412235 0.7336046
[31,] 0.8449947 1.2419029 0.70222505 0.72985869 1.1084839 1.5853875 0.9939843
[32,] 0.8940978 1.4739752 0.40913348 0.46675006 0.4452300 1.1734321 0.3421238
[33,] 1.6781560 2.1788496 1.34266001 1.35324741 0.8716884 1.1163193 0.9797778
[34,] 1.3287062 1.8713363 0.94070004 0.95983172 0.4674725 0.9743940 0.5655832
[35,] 0.8072681 1.3318713 0.45430815 0.50403791 0.7651139 1.3608638 0.6502654
[36,] 1.3325550 1.8740163 0.94657291 0.96528902 0.4730484 0.9719066 0.5719383
[37,] 2.9221351 3.2923606 2.70585720 2.70088549 2.2654506 2.1066588 2.3800282
[38,] 4.4563002 4.7828638 4.26025143 4.25397549 3.8208256 3.6115931 3.9352853
[,8] [,9] [,10] [,11] [,12] [,13] [,14]
[1,] 1.0768523 1.38574578 1.0787200 1.5785857 4.9582430 4.3920042 0.4463863
[2,] 1.3062580 1.92675597 1.4327013 1.6203681 4.4677868 3.9715908 0.9188558
[3,] 1.1433473 0.99785275 0.9351673 1.6699910 5.3997500 4.7920987 0.3986693
[4,] 1.1003214 1.01729611 0.9664700 1.6792311 5.3402801 4.7329359 0.3993572
[5,] 1.0091925 0.52489553 1.3157096 2.1163496 5.5574301 4.8810736 0.8690649
[6,] 0.2341955 1.01279559 1.8261371 2.4477769 4.8146915 4.1039300 1.2047738
[7,] 1.0772403 0.62215440 1.2005513 2.0040094 5.5747362 4.9145198 0.7661213
[8,] 0.0000000 1.15345243 2.0555505 2.6546328 4.6674438 3.9295873 1.4285194
[9,] 1.1534524 0.00000000 1.7685665 2.6068844 5.8187843 5.0792800 1.3882319
[10,] 2.0555505 1.76856653 0.0000000 0.8790767 5.8906104 5.4015022 0.6433198
[11,] 2.6546328 2.60688442 0.8790767 0.0000000 5.7966350 5.4500514 1.2818624
[12,] 4.6674438 5.81878427 5.8906104 5.7966350 0.0000000 1.0048121 5.3708188
[13,] 3.9295873 5.07927997 5.4015022 5.4500514 1.0048121 0.0000000 4.8276673
[14,] 1.4285194 1.38823188 0.6433198 1.2818624 5.3708188 4.8276673 0.0000000
[15,] 4.0134118 5.16623718 5.4133958 5.4244002 0.7963007 0.2318194 4.8532770
[16,] 0.7432475 1.18339530 1.3486550 1.9116823 4.8821683 4.2596756 0.7068299
[17,] 1.3750919 0.99995667 0.7891324 1.6070305 5.6591066 5.0572864 0.4528387
[18,] 0.5139306 0.91002115 2.2748044 2.9777096 5.0389149 4.2453801 1.7017272
[19,] 0.2805119 1.22652613 2.3174701 2.9335105 4.6534054 3.8726343 1.6981145
[20,] 0.8911399 0.59903125 2.2405058 3.0219802 5.4768274 4.6794261 1.7460374
[21,] 1.1213294 0.40196737 1.3767956 2.2062667 5.7089986 5.0209844 0.9913216
[22,] 1.0039977 0.29201421 1.5350921 2.3502022 5.6341117 4.9255513 1.1086010
[23,] 1.6889235 1.38376529 0.3984978 1.2279119 5.7383531 5.1940741 0.3682217
[24,] 1.4333046 0.83135905 0.9415162 1.7991006 5.8472512 5.2185503 0.7060818
[25,] 1.5643432 1.23969360 0.5455014 1.3684769 5.7013818 5.1356056 0.3502300
[26,] 0.9697679 0.67978067 1.2247731 1.9963209 5.4468623 4.7884976 0.7307428
[27,] 0.4631490 1.02742561 1.5935411 2.2043055 4.8698704 4.1967899 0.9655393
[28,] 1.2187969 0.93491618 0.9090095 1.6903626 5.5276188 4.9142735 0.4568654
[29,] 1.0199549 0.13870864 1.8013957 2.6213035 5.6869468 4.9430603 1.3769066
[30,] 1.0502428 0.18753965 1.9147603 2.7365918 5.7152888 4.9571476 1.4916455
[31,] 1.8142108 1.58559958 0.2420736 1.0229398 5.7094272 5.1974032 0.4037650
[32,] 1.4023327 0.84721471 0.9215429 1.7716694 5.7992147 5.1737987 0.6585564
[33,] 1.1862334 0.38475233 2.1503301 2.9802696 5.8214017 5.0356071 1.7399755
[34,] 1.1230313 0.05758741 1.7188514 2.5538833 5.7849464 5.0515753 1.3317036
[35,] 1.5950050 1.21692561 0.5544840 1.3981877 5.7644510 5.1941306 0.4180266
[36,] 1.1190800 0.05319223 1.7269595 2.5614037 5.7817082 5.0472962 1.3380300
[37,] 2.0058761 1.82808249 3.5765441 4.3748634 5.9220546 4.9989142 3.1002107
[38,] 3.4770940 3.37231092 5.1301564 5.9298467 6.6530866 5.6618770 4.6533043
[,15] [,16] [,17] [,18] [,19] [,20] [,21]
[1,] 4.4229974 0.3370591 0.7190392 1.4427677 1.3565843 1.5913255 1.0403113
[2,] 3.9808059 0.7519071 1.2891703 1.7785758 1.5707411 2.0274812 1.6092336
[3,] 4.8365921 0.5344246 0.2657686 1.3416789 1.3932763 1.3522049 0.6095113
[4,] 4.7771803 0.4761288 0.3245247 1.3173841 1.3545336 1.3467192 0.6370364
[5,] 4.9470266 0.7350235 0.5265886 1.0224024 1.1941739 0.9250033 0.1682477
[6,] 4.1791099 0.5410441 1.1412164 0.5937101 0.4933590 0.8774601 0.9211032
[7,] 4.9754554 0.7166140 0.4114340 1.1261601 1.2777324 1.0401161 0.2314572
[8,] 4.0134118 0.7432475 1.3750919 0.5139306 0.2805119 0.8911399 1.1213294
[9,] 5.1662372 1.1833953 0.9999567 0.9100212 1.2265261 0.5990312 0.4019674
[10,] 5.4133958 1.3486550 0.7891324 2.2748044 2.3174701 2.2405058 1.3767956
[11,] 5.4244002 1.9116823 1.6070305 2.9777096 2.9335105 3.0219802 2.2062667
[12,] 0.7963007 4.8821683 5.6591066 5.0389149 4.6534054 5.4768274 5.7089986
[13,] 0.2318194 4.2596756 5.0572864 4.2453801 3.8726343 4.6794261 5.0209844
[14,] 4.8532770 0.7068299 0.4528387 1.7017272 1.6981145 1.7460374 0.9913216
[15,] 0.0000000 4.3071466 5.1006906 4.3495762 3.9713660 4.7858283 5.0909532
[16,] 4.3071466 0.0000000 0.8001766 1.1121545 1.0218345 1.2951354 0.9028708
[17,] 5.1006906 0.8001766 0.0000000 1.5116077 1.6105253 1.4515067 0.5993698
[18,] 4.3495762 1.1121545 1.5116077 0.0000000 0.3907137 0.4379209 1.0618937
[19,] 3.9713660 1.0218345 1.6105253 0.3907137 0.0000000 0.8262964 1.2801150
[20,] 4.7858283 1.2951354 1.4515067 0.4379209 0.8262964 0.0000000 0.8924249
[21,] 5.0909532 0.9028708 0.5993698 1.0618937 1.2801150 0.8924249 0.0000000
[22,] 5.0019718 0.9057144 0.7493099 0.8974563 1.1385643 0.7205007 0.1734859
[23,] 5.2209729 1.0172497 0.3921312 1.8815404 1.9427706 1.8425264 0.9867489
[24,] 5.2704240 0.9657572 0.2540200 1.4813754 1.6403399 1.3489650 0.4572878
[25,] 5.1685695 0.9187779 0.2447468 1.7399684 1.8128736 1.6951603 0.8412099
[26,] 4.8485608 0.5923575 0.4482042 1.0635299 1.1834458 1.0272045 0.3235515
[27,] 4.2599423 0.3015187 0.9426241 0.8127404 0.7342846 1.0294122 0.8399194
[28,] 4.9607941 0.6546053 0.1572551 1.3716717 1.4569709 1.3399660 0.5354337
[29,] 5.0312510 1.1031253 1.0186146 0.7741697 1.0878877 0.4916068 0.4255309
[30,] 5.0502176 1.1971583 1.1332747 0.7479727 1.0884956 0.4118657 0.5380584
[31,] 5.2151649 1.1072012 0.5877900 2.0438986 2.0773131 2.0291439 1.1865383
[32,] 5.2245771 0.9186276 0.2057326 1.4668195 1.6146980 1.3497145 0.4623685
[33,] 5.1384352 1.4272720 1.3744450 0.7903391 1.1686300 0.3644679 0.7756301
[34,] 5.1364296 1.1293904 0.9468592 0.9042206 1.2080079 0.6178737 0.3478188
[35,] 5.2285590 0.9666951 0.2415724 1.7517719 1.8386439 1.6898193 0.8226330
[36,] 5.1324905 1.1313530 0.9544023 0.8970462 1.2023070 0.6090385 0.3552109
[37,] 5.1560502 2.6053301 2.7894362 1.5137976 1.7809769 1.3541737 2.2025326
[38,] 5.8574573 4.1311861 4.3441246 3.0212588 3.2212781 2.9080492 3.7542985
[,22] [,23] [,24] [,25] [,26] [,27] [,28]
[1,] 1.0944649 0.8047850 0.94587078 0.74592497 0.7186464 0.6373845 0.6176946
[2,] 1.6415196 1.2799855 1.52586950 1.26507259 1.2857274 0.9984186 1.2002957
[3,] 0.7133470 0.5500913 0.46028644 0.42100477 0.3320741 0.6949768 0.1306694
[4,] 0.7290357 0.5889085 0.51757881 0.46624106 0.3411891 0.6472487 0.1888638
[5,] 0.2395496 0.9181612 0.45920405 0.77069092 0.1601017 0.6882807 0.4262828
[6,] 0.8228824 1.4562765 1.20608113 1.33061450 0.7425485 0.2435020 0.9847711
[7,] 0.3470134 0.8030365 0.36313265 0.65556064 0.1279154 0.7182704 0.3159500
[8,] 1.0039977 1.6889235 1.43330459 1.56434316 0.9697679 0.4631490 1.2187969
[9,] 0.2920142 1.3837653 0.83135905 1.23969360 0.6797807 1.0274256 0.9349162
[10,] 1.5350921 0.3984978 0.94151617 0.54550136 1.2247731 1.5935411 0.9090095
[11,] 2.3502022 1.2279119 1.79910059 1.36847695 1.9963209 2.2043055 1.6903626
[12,] 5.6341117 5.7383531 5.84725119 5.70138184 5.4468623 4.8698704 5.5276188
[13,] 4.9255513 5.1940741 5.21855026 5.13560559 4.7884976 4.1967899 4.9142735
[14,] 1.1086010 0.3682217 0.70608176 0.35022996 0.7307428 0.9655393 0.4568654
[15,] 5.0019718 5.2209729 5.27042404 5.16856945 4.8485608 4.2599423 4.9607941
[16,] 0.9057144 1.0172497 0.96575716 0.91877793 0.5923575 0.3015187 0.6546053
[17,] 0.7493099 0.3921312 0.25402002 0.24474679 0.4482042 0.9426241 0.1572551
[18,] 0.8974563 1.8815404 1.48137543 1.73996841 1.0635299 0.8127404 1.3716717
[19,] 1.1385643 1.9427706 1.64033995 1.81287357 1.1834458 0.7342846 1.4569709
[20,] 0.7205007 1.8425264 1.34896496 1.69516030 1.0272045 1.0294122 1.3399660
[21,] 0.1734859 0.9867489 0.45728777 0.84120988 0.3235515 0.8399194 0.5354337
[22,] 0.0000000 1.1408102 0.62848475 0.99380069 0.3892264 0.7853218 0.6628539
[23,] 1.1408102 0.0000000 0.57254249 0.14748221 0.8269052 1.2325081 0.5117531
[24,] 0.6284847 0.5725425 0.00000000 0.44109923 0.4637018 1.0435629 0.3297361
[25,] 0.9938007 0.1474822 0.44109923 0.00000000 0.6817912 1.1134889 0.3683614
[26,] 0.3892264 0.8269052 0.46370178 0.68179117 0.0000000 0.5936322 0.3160615
[27,] 0.7853218 1.2325081 1.04356292 1.11348890 0.5936322 0.0000000 0.7855328
[28,] 0.6628539 0.5117531 0.32973612 0.36836140 0.3160615 0.7855328 0.0000000
[29,] 0.2712388 1.4091246 0.87924737 1.26256780 0.6501411 0.9227757 0.9338779
[30,] 0.3866451 1.5233322 0.98860086 1.37697468 0.7637266 0.9995892 1.0492873
[31,] 1.3363915 0.2061907 0.77871388 0.34596067 1.0050737 1.3519026 0.6896898
[32,] 0.6300709 0.5437640 0.05189479 0.40712075 0.4331245 1.0048375 0.2784886
[33,] 0.6329888 1.7622751 1.21558314 1.61683616 1.0124258 1.2088059 1.2957211
[34,] 0.2346134 1.3322648 0.78433147 1.18760261 0.6221936 0.9797841 0.8790567
[35,] 0.9808281 0.1707486 0.40179571 0.06788136 0.6890472 1.1492840 0.3836987
[36,] 0.2389824 1.3401063 0.79279208 1.19534165 0.6272010 0.9795185 0.8857468
[37,] 2.0414535 3.1815046 2.65405172 3.03418260 2.3786242 2.3141705 2.6879184
[38,] 3.5953907 4.7360437 4.20239792 4.58883788 3.9338748 3.8335698 4.2435113
[,29] [,30] [,31] [,32] [,33] [,34] [,35]
[1,] 1.3306488 1.4357947 0.8449947 0.89409778 1.6781560 1.328706174 0.80726808
[2,] 1.8530637 1.9487727 1.2419029 1.47397521 2.1788496 1.871336322 1.33187135
[3,] 0.9797193 1.0940889 0.7022251 0.40913348 1.3426600 0.940700045 0.45430815
[4,] 0.9912643 1.1045800 0.7298587 0.46675006 1.3532474 0.959831723 0.50403791
[5,] 0.5088735 0.6241097 1.1084839 0.44522998 0.8716884 0.467472530 0.76511391
[6,] 0.8887040 0.9412235 1.5853875 1.17343214 1.1163193 0.974393976 1.36086384
[7,] 0.6181842 0.7336046 0.9939843 0.34212378 0.9797778 0.565583151 0.65026542
[8,] 1.0199549 1.0502428 1.8142108 1.40233266 1.1862334 1.123031288 1.59500504
[9,] 0.1387086 0.1875397 1.5855996 0.84721471 0.3847523 0.057587409 1.21692561
[10,] 1.8013957 1.9147603 0.2420736 0.92154288 2.1503301 1.718851391 0.55448404
[11,] 2.6213035 2.7365918 1.0229398 1.77166937 2.9802696 2.553883280 1.39818768
[12,] 5.6869468 5.7152888 5.7094272 5.79921471 5.8214017 5.784946440 5.76445105
[13,] 4.9430603 4.9571476 5.1974032 5.17379866 5.0356071 5.051575266 5.19413058
[14,] 1.3769066 1.4916455 0.4037650 0.65855639 1.7399755 1.331703642 0.41802663
[15,] 5.0312510 5.0502176 5.2151649 5.22457713 5.1384352 5.136429567 5.22855899
[16,] 1.1031253 1.1971583 1.1072012 0.91862761 1.4272720 1.129390421 0.96669511
[17,] 1.0186146 1.1332747 0.5877900 0.20573262 1.3744450 0.946859219 0.24157240
[18,] 0.7741697 0.7479727 2.0438986 1.46681952 0.7903391 0.904220627 1.75177187
[19,] 1.0878877 1.0884956 2.0773131 1.61469795 1.1686300 1.208007902 1.83864386
[20,] 0.4916068 0.4118657 2.0291439 1.34971446 0.3644679 0.617873706 1.68981926
[21,] 0.4255309 0.5380584 1.1865383 0.46236850 0.7756301 0.347818753 0.82263296
[22,] 0.2712388 0.3866451 1.3363915 0.63007090 0.6329888 0.234613443 0.98082807
[23,] 1.4091246 1.5233322 0.2061907 0.54376396 1.7622751 1.332264763 0.17074858
[24,] 0.8792474 0.9886009 0.7787139 0.05189479 1.2155831 0.784331469 0.40179571
[25,] 1.2625678 1.3769747 0.3459607 0.40712075 1.6168362 1.187602615 0.06788136
[26,] 0.6501411 0.7637266 1.0050737 0.43312448 1.0124258 0.622193630 0.68904720
[27,] 0.9227757 0.9995892 1.3519026 1.00483748 1.2088059 0.979784143 1.14928403
[28,] 0.9338779 1.0492873 0.6896898 0.27848859 1.2957211 0.879056734 0.38369867
[29,] 0.0000000 0.1154206 1.6062524 0.88739164 0.3630692 0.132419371 1.24694032
[30,] 0.1154206 0.0000000 1.7210135 0.99859586 0.2486994 0.213783207 1.36043447
[31,] 1.6062524 1.7210135 0.0000000 0.74930307 1.9620189 1.533345245 0.37692096
[32,] 0.8873916 0.9985959 0.7493031 0.00000000 1.2299902 0.798247942 0.37368812
[33,] 0.3630692 0.2486994 1.9620189 1.22999020 0.0000000 0.431744190 1.59716464
[34,] 0.1324194 0.2137832 1.5333452 0.79824794 0.4317442 0.000000000 1.16621930
[35,] 1.2469403 1.3604345 0.3769210 0.37368812 1.5971646 1.166219297 0.00000000
[36,] 0.1244578 0.2047653 1.5410399 0.80649534 0.4235072 0.009019522 1.17419961
[37,] 1.7770094 1.6657155 3.3739334 2.66415955 1.4446632 1.871301031 3.02224241
[38,] 3.3290215 3.2163804 4.9292618 4.21460029 2.9875592 3.418100407 4.57568091
[,36] [,37] [,38]
[1,] 1.332555020 2.922135 4.456300
[2,] 1.874016322 3.292361 4.782864
[3,] 0.946572908 2.705857 4.260251
[4,] 0.965289018 2.700885 4.253975
[5,] 0.473048382 2.265451 3.820826
[6,] 0.971906559 2.106659 3.611593
[7,] 0.571938338 2.380028 3.935285
[8,] 1.119079964 2.005876 3.477094
[9,] 0.053192229 1.828082 3.372311
[10,] 1.726959463 3.576544 5.130156
[11,] 2.561403721 4.374863 5.929847
[12,] 5.781708222 5.922055 6.653087
[13,] 5.047296167 4.998914 5.661877
[14,] 1.338030035 3.100211 4.653304
[15,] 5.132490525 5.156050 5.857457
[16,] 1.131353028 2.605330 4.131186
[17,] 0.954402283 2.789436 4.344125
[18,] 0.897046230 1.513798 3.021259
[19,] 1.202307010 1.780977 3.221278
[20,] 0.609038547 1.354174 2.908049
[21,] 0.355210941 2.202533 3.754298
[22,] 0.238982394 2.041454 3.595391
[23,] 1.340106253 3.181505 4.736044
[24,] 0.792792075 2.654052 4.202398
[25,] 1.195341654 3.034183 4.588838
[26,] 0.627201028 2.378624 3.933875
[27,] 0.979518532 2.314170 3.833570
[28,] 0.885746844 2.687918 4.243511
[29,] 0.124457797 1.777009 3.329022
[30,] 0.204765302 1.665715 3.216380
[31,] 1.541039857 3.373933 4.929262
[32,] 0.806495343 2.664160 4.214600
[33,] 0.423507203 1.444663 2.987559
[34,] 0.009019522 1.871301 3.418100
[35,] 1.174199615 3.022242 4.575681
[36,] 0.000000000 1.862635 3.409615
[37,] 1.862635025 0.000000 1.555594
[38,] 3.409615425 1.555594 0.000000Output menunjukkan nilai-nilai jarak rekonstruksi antar sejumlah objek dalam format matriks terformat. Nilai-nilai yang terlihat berkisar dari dekat nol sampai beberapa unit skala, yang mencerminkan perbedaan relatif antar provinsi dalam ruang hasil MDS. Diagonal matriks bernilai nol karena jarak objek terhadap dirinya sendiri adalah nol, dan bahwa elemen-elemen di luar diagonal memberikan ukuran seberapa jauh pasangan objek terpisah setelah embedding. Informasi numerik semacam ini diperlukan untuk menghitung ukuran kesalahan rekonstruksi seperti STRESS dan untuk membandingkan jarak rekonstruksi dengan jarak asli yang dihitung dari data terstandardisasi.
4.6 Nilai STRESS
> stress <- sqrt(sum((D - disparities)^2) / sum(D^2))
> cat("Nilai Stress:", stress, "\n")
Nilai Stress: 0.008789678 Output MDS ini menampilkan nilai STRESS sebesar 0.008789678. Nilai STRESS yang sangat kecil menunjukkan bahwa konfigurasi dua dimensi merekonstruksi jarak asli dengan sangat baik dan bahwa representasi 2 dimensi dapat dipercaya untuk menggambarkan struktur kemiripan antar provinsi. Dalam interpretasi metodologis, nilai STRESS jauh di bawah ambang umum 0,05 digolongkan sebagai fit yang sangat baik sehingga visualisasi dapat dijadikan dasar untuk analisis deskriptif dan identifikasi pola.
5 KESIMPULAN
Berdasarkan hasil Multidimensional Scaling dua dimensi, pola kemiripan antar provinsi menurut variabel Jumlah Keluarga Penerima Manfaat (KPM), jumlah penduduk miskin, dan rata-rata lama sekolah (RLS) menunjukkan beberapa temuan penting. Konfigurasi dua dimensi menangkap hampir seluruh variasi data dengan kontribusi Dimensi 1 sekitar 66,84 persen dan Dimensi 1 ditambah Dimensi 2 sekitar 99,47 persen serta nilai STRESS sebesar 0,00879 yang menandakan rekonstruksi jarak yang sangat baik. Sebagian besar provinsi membentuk satu kelompok yang relatif homogen yang mencerminkan profil kombinasi KPM, tingkat kemiskinan, dan RLS yang serupa. Terdapat sejumlah provinsi yang terpisah dari kelompok mayor sehingga menunjukkan perbedaan profil yang konsisten. Beberapa provinsi muncul sebagai outlier dengan posisi menonjol pada salah satu dimensi yang mengindikasikan karakteristik ekstrem atau unik pada variabel yang dianalisis. Temuan ini mengindikasikan bahwa meskipun banyak provinsi menunjukkan kemiripan sosial ekonomi, terdapat pula kelompok dan kasus khusus yang perlu dikaji lebih lanjut dengan melihat data asli dan konteks regional.
6 DAFTAR PUSTAKA
Badan Pusat Statistik Indonesia. Jumlah Keluarga Penerima Manfaat (KPM) dan Anggaran Bantuan Sosial Pangan Menurut Provinsi, 2024. Diakses pada 18 November 2025, dari https://www.bps.go.id/id/statistics-table/3/TWt0MVNGZFdiV2RaYTFoS1oyRnRSVTFOYUhSc1VUMDkjMyMwMDAw/jumlah-keluarga-penerima-manfaat--kpm--dan-anggaran-bantuan-sosial-pangan-menurut-provinsi.html?year=2024
Badan Pusat Statistik Indonesia. (2 September 2024). Jumlah Penduduk Miskin (Ribu Jiwa) Menurut Kabupaten/Kota , 2024. Diakses pada 21 November 2025, dari https://www.bps.go.id/id/statistics-table/2/NjE5IzI=/jumlah-penduduk-miskin--ribu-jiwa--menurut-kabupaten-kota-.html
Badan Pusat Statistik Indonesia. (5 November 2025). [Metode Baru] Rata-rata Lama Sekolah, 2025. Diakses pada 21 November 2025, dari https://www.bps.go.id/id/statistics-table/2/NDE1IzI=/-metode-baru--rata-rata-lama-sekolah--tahun-.html
Hair, J. F., Black, W. C., Babin, B. J., & Anderson, R. E. (2019). Multivariate data analysis (8th ed.). Cengage Learning.
Johnson, R. A., & Wichern, D. W. (2007). Applied multivariate statistical analysis (6th ed.). Pearson.
Kruskal, J. B., & Wish, M. (1978). Multidimensional scaling. Sage Publications.
Sugiyono. (2019). Metode penelitian kuantitatif, kualitatif, dan R&D. Alfabeta.