1 Studi Kasus

1.1 Data

1.1.1 Sumber Data

Dataset USArrests terdiri dari empat variabel utama yang menggambarkan kondisi kriminalitas dan karakteristik kependudukan di 50 negara bagian Amerika Serikat pada tahun 1973. Dataset USArrests berasal dari publikasi resmi pemerintah Amerika Serikat berjudul Statistical Abstract of the United States 1975 yang diterbitkan oleh U.S. Department of Commerce, Bureau of the Census. Publikasi tersebut memuat ringkasan statistik nasional, termasuk data kriminalitas tahun 1973 yang kemudian dirangkum kembali menjadi empat indikator utama: Murder, Assault, UrbanPop, dan Rape untuk 50 negara bagian Amerika Serikat. Dataset ini kemudian dikurasi dan disertakan dalam paket datasets pada perangkat lunak R, sehingga dapat digunakan secara luas dalam penelitian statistik dan analisis multivariat.

1.1.2 Studi Kasus dari Data

Analisis ini bertujuan untuk mengetahui pola kemiripan tingkat kriminalitas antar negara bagian di Amerika Serikat dengan menggunakan data USArrests, yang memuat empat indikator utama kejahatan pada tahun 1973, yaitu Murder, Assault, UrbanPop, dan Rape. Melalui analisis Multidimensional Scaling (MDS), penelitian ini berupaya memetakan 50 negara bagian ke dalam ruang berdimensi rendah sehingga hubungan kedekatan atau perbedaan karakteristik kriminalitas antar wilayah dapat divisualisasikan secara lebih jelas. Pendekatan ini memungkinkan untuk mengidentifikasi kelompok negara bagian yang memiliki pola kriminalitas serupa, serta mendeteksi negara bagian yang menunjukkan karakteristik yang berbeda secara signifikan dari wilayah lainnya. Dengan demikian, MDS memberikan gambaran komprehensif mengenai persebaran tingkat kejahatan di Amerika Serikat yang dapat mendukung pemahaman lebih dalam mengenai faktor-faktor yang mungkin mempengaruhi kriminalitas di berbagai wilayah.

1.1.3 Deskripsi Data

> library(knitr)
> library(readxl)
> tabel <- read_excel("D:/MATERI KULIAH/SMT 5/data_anmul.xlsx")
> tabel
# A tibble: 50 × 5
   State       Murder Assault UrbanPop  Rape
   <chr>        <dbl>   <dbl>    <dbl> <dbl>
 1 Alabama       13.2     236       58  21.2
 2 Alaska        10       263       48  44.5
 3 Arizona        8.1     294       80  31  
 4 Arkansas       8.8     190       50  19.5
 5 California     9       276       91  40.6
 6 Colorado       7.9     204       78  38.7
 7 Connecticut    3.3     110       77  11.1
 8 Delaware       5.9     238       72  15.8
 9 Florida       15.4     335       80  31.9
10 Georgia       17.4     211       60  25.8
# ℹ 40 more rows
> knitr::kable(tabel,caption = "Tabel Tingkat Kriminalitas Negara Bagian Amerika Serikat Tahun 1973")
Tabel Tingkat Kriminalitas Negara Bagian Amerika Serikat Tahun 1973
State Murder Assault UrbanPop Rape
Alabama 13.2 236 58 21.2
Alaska 10.0 263 48 44.5
Arizona 8.1 294 80 31.0
Arkansas 8.8 190 50 19.5
California 9.0 276 91 40.6
Colorado 7.9 204 78 38.7
Connecticut 3.3 110 77 11.1
Delaware 5.9 238 72 15.8
Florida 15.4 335 80 31.9
Georgia 17.4 211 60 25.8
Hawaii 5.3 46 83 20.2
Idaho 2.6 120 54 14.2
Illinois 10.4 249 83 24.0
Indiana 7.2 113 65 21.0
Iowa 2.2 56 57 11.3
Kansas 6.0 115 66 18.0
Kentucky 9.7 109 52 16.3
Louisiana 15.4 249 66 22.2
Maine 2.1 83 51 7.8
Maryland 11.3 300 67 27.8
Massachusetts 4.4 149 85 16.3
Michigan 12.1 255 74 35.1
Minnesota 2.7 72 66 14.9
Mississippi 16.1 259 44 17.1
Missouri 9.0 178 70 28.2
Montana 6.0 109 53 16.4
Nebraska 4.3 102 62 16.5
Nevada 12.2 252 81 46.0
New Hampshire 2.1 57 56 9.5
New Jersey 7.4 159 89 18.8
New Mexico 11.4 285 70 32.1
New York 11.1 254 86 26.1
North Carolina 13.0 337 45 16.1
North Dakota 0.8 45 44 7.3
Ohio 7.3 120 75 21.4
Oklahoma 6.6 151 68 20.0
Oregon 4.9 159 67 29.3
Pennsylvania 6.3 106 72 14.9
Rhode Island 3.4 174 87 8.3
South Carolina 14.4 279 48 22.5
South Dakota 3.8 86 45 12.8
Tennessee 13.2 188 59 26.9
Texas 12.7 201 80 25.5
Utah 3.2 120 80 22.9
Vermont 2.2 48 32 11.2
Virginia 8.5 156 63 20.7
Washington 4.0 145 73 26.2
West Virginia 5.7 81 39 9.3
Wisconsin 2.6 53 66 10.8
Wyoming 6.8 161 60 15.6

Variabel Murder menyatakan jumlah penangkapan terkait kasus pembunuhan per 100.000 penduduk, sehingga mencerminkan tingkat kejahatan berat yang paling ekstrem. Variabel Assault menggambarkan jumlah penangkapan untuk tindak penyerangan per 100.000 penduduk, yang umumnya memiliki angka lebih tinggi dibandingkan pembunuhan sehingga menjadi salah satu indikator penting dalam memetakan tingkat kekerasan di suatu wilayah. Variabel UrbanPop menunjukkan persentase penduduk yang tinggal di wilayah urban, dan digunakan untuk menggambarkan tingkat urbanisasi tiap negara bagian, karena urbanisasi sering dianggap berhubungan dengan dinamika kriminalitas. Terakhir, variabel Rape berisi jumlah penangkapan terkait kasus pemerkosaan per 100.000 penduduk, yang juga menjadi indikator penting dalam analisis kejahatan berbasis populasi. Keempat variabel ini bersama-sama memberikan gambaran menyeluruh mengenai pola kriminalitas dan demografi di Amerika Serikat, sehingga dataset ini banyak digunakan dalam analisis multivariat seperti PCA, klaster, maupun Multidimensional Scaling (MDS) untuk memahami kemiripan dan perbedaan karakteristik antar negara bagian.

1.2 Latar Belakang Metode

Analisis Multidimensional Scaling (MDS) merupakan metode statistik multivariat yang digunakan untuk memetakan objek-objek berdasarkan tingkat kemiripannya ke dalam ruang berdimensi rendah, biasanya dua atau tiga dimensi, sehingga hubungan antar objek dapat divisualisasikan secara lebih intuitif. Metode ini berangkat dari asumsi bahwa informasi mengenai kemiripan atau jarak antar objek dapat direpresentasikan dalam bentuk koordinat geometris yang mempertahankan struktur kedekatannya. MDS banyak digunakan dalam berbagai bidang seperti psikologi, pemasaran, geografi, hingga analisis sosial karena kemampuannya menyederhanakan data kompleks menjadi peta visual yang mudah ditafsirkan. Dalam konteks penelitian ini, MDS dipilih karena mampu menggambarkan pola kedekatan antar negara bagian Amerika Serikat berdasarkan indikator kriminalitas secara komprehensif, sehingga memudahkan identifikasi kelompok wilayah dengan karakteristik kejahatan yang serupa maupun yang berbeda secara signifikan. Keunggulan utama MDS terletak pada fleksibilitasnya dalam mengolah data berbasis jarak dan kemampuannya menghasilkan visualisasi yang dapat mendukung interpretasi yang lebih mendalam terhadap struktur data.

1.3 Tinjauan Pustaka

1.3.1 Data Kriminalitas dan Analisis Statistik

Data kriminalitas sering dianalisis untuk memahami pola kejadian kejahatan di suatu wilayah dan karakteristik sosial yang mungkin memengaruhinya. Berbagai indikator seperti tingkat pembunuhan (Murder), penyerangan (Assault), dan proporsi populasi perkotaan (UrbanPop) digunakan untuk menggambarkan kondisi keamanan suatu wilayah. Analisis statistik terhadap data kriminalitas membantu dalam mengidentifikasi wilayah dengan risiko tinggi, memahami hubungan antar jenis kejahatan, serta menyediakan dasar empiris bagi perumusan kebijakan penanggulangan kriminalitas. Studi-studi sebelumnya menunjukkan bahwa variasi geografis dan faktor demografis sering memengaruhi tingkat kejahatan di berbagai negara bagian (FBI, 1975; Osgood, 2000).

1.3.2 Analisis Multidimensional Scaling (MDS)

Multidimensional Scaling (MDS) adalah metode statistik multivariat yang digunakan untuk memetakan objek ke dalam ruang berdimensi rendah berdasarkan tingkat kemiripan atau jarak antar objek. MDS bertujuan untuk menghasilkan representasi visual yang mempertahankan struktur jarak pada data asli sehingga pola-pola seperti kelompok, kedekatan, atau perbedaan antar objek dapat diamati dengan lebih mudah. Dalam konteks data kriminalitas, MDS memungkinkan peneliti untuk melihat bagaimana negara bagian saling berhubungan berdasarkan tingkat kejahatan yang terjadi, serta mengidentifikasi kelompok wilayah yang menunjukkan karakteristik kriminalitas serupa. MDS telah digunakan secara luas pada berbagai bidang seperti psikologi, pemasaran, biologi, dan kriminologi.

Tahapan analisis Multidimensional Scaling (MDS) adalah sebagai berikut :

  1. Tahap pertama dalam analisis MDS adalah menyiapkan data dan menghitung matriks jarak antar objek. Data yang digunakan biasanya bersifat multivariat numerik sehingga setiap objek memiliki beberapa variabel pengukuran. Sebelum menghitung jarak, data umumnya distandardisasi untuk menghilangkan pengaruh perbedaan skala antar variabel. Setelah itu, jarak antar objek dihitung menggunakan ukuran kedekatan tertentu, seperti jarak Euclidean, Manhattan, atau jarak lainnya sesuai kebutuhan penelitian. Hasil perhitungan tersebut berupa matriks jarak yang akan menjadi dasar proses pemetaan MDS.

  2. Tahap berikutnya adalah mengkonversi matriks jarak menjadi konfigurasi titik dalam ruang berdimensi rendah. Pada tahap ini, metode MDS bekerja dengan mencari koordinat yang paling baik merepresentasikan jarak antar objek sedekat mungkin dengan jarak asli dalam matriks. Algoritma MDS berusaha meminimalkan perbedaan antara jarak asli dan jarak yang dihasilkan dari konfigurasi posisi objek dalam ruang dua atau tiga dimensi. Pada MDS klasik, proses ini melibatkan dekomposisi eigen dari matriks yang telah ditransformasi menjadi bentuk matriks kesamaan (inner product matrix). Sedangkan pada non-metric MDS, transformasi dilakukan menggunakan fungsi monotonik untuk mempertahankan urutan peringkat jarak.

  3. Tahap ketiga adalah evaluasi kualitas konfigurasi menggunakan ukuran stress. Stress merupakan ukuran ketidaksesuaian antara jarak asli dan jarak hasil pemetaan. Nilai stress yang rendah menunjukkan representasi konfigurasi yang baik, sedangkan nilai yang tinggi menunjukkan bahwa ruang berdimensi rendah tidak cukup menggambarkan struktur jarak data sebenarnya. Nilai stress menjadi indikator penting untuk menentukan apakah jumlah dimensi yang dipilih sudah tepat atau perlu ditambah atau dikurangi.

  4. Tahap terakhir adalah interpretasi hasil dan visualisasi konfigurasi MDS. Hasil pemetaan biasanya ditampilkan dalam bentuk scatter plot dua dimensi yang memuat objek-objek berdasarkan kedekatannya. Objek yang berdekatan dalam grafik menunjukkan tingkat kemiripan yang tinggi, sedangkan yang berjauhan menunjukkan perbedaan karakteristik yang besar. Visualisasi ini memudahkan peneliti untuk mengidentifikasi pola, kelompok (cluster), atau outlier dalam data. Interpretasi dapat diperkuat dengan membandingkan posisi objek dengan karakteristik variabel aslinya.

1.3.3 Penggunaan Jarak Euclidean dalam Analisis Multivariat

Jarak Euclidean merupakan ukuran jarak yang paling umum digunakan dalam analisis multivariat, termasuk dalam MDS. Ukuran jarak ini menghitung kedekatan antar objek berdasarkan perbedaan nilai dari seluruh variabel numerik. Dalam data USArrests, penggunaan jarak Euclidean memungkinkan peneliti untuk menilai seberapa mirip dua negara bagian berdasarkan tingkat kejahatan dan komposisi penduduk urban. Literatur menunjukkan bahwa jarak Euclidean sangat sensitif terhadap skala variabel sehingga normalisasi atau standardisasi sering diperlukan terutama ketika variabel memiliki satuan yang berbeda. Oleh karena itu, penggunaan jarak Euclidean yang telah distandardisasi menjadi pendekatan umum dalam analisis seperti MDS (Everitt et al., 2011; Johnson & Wichern, 2007).

1.3.4 Visualisasi Data untuk Pemahaman Pola Kriminalitas

Visualisasi data merupakan komponen penting dalam analisis statistik modern karena memudahkan interpretasi dan komunikasi hasil. Dalam penelitian kriminalitas, visualisasi seperti plot MDS membantu dalam mengidentifikasi pola kedekatan, outlier, atau kelompok wilayah dengan karakteristik serupa. Literatur menyatakan bahwa representasi grafis mampu menyampaikan informasi kompleks dengan lebih intuitif dibandingkan analisis numerik semata. Oleh karena itu, penggabungan metode visual seperti MDS dengan data kriminalitas memungkinkan peneliti menghasilkan pemahaman yang lebih holistik terhadap fenomena yang diteliti (Cleveland, 1993; Friendly, 2008).

1.4 Tujuan Penelitian

Penelitian ini bertujuan untuk menganalisis pola kemiripan dan perbedaan tingkat kriminalitas antar negara bagian di Amerika Serikat berdasarkan dataset USArrests. Secara khusus, penelitian ini bertujuan untuk memetakan 50 negara bagian ke dalam ruang berdimensi rendah menggunakan metode Multidimensional Scaling (MDS) agar struktur jarak antar negara bagian dapat divisualisasikan secara lebih jelas. Selain itu, penelitian ini bertujuan untuk mengidentifikasi kelompok negara bagian yang memiliki karakteristik kriminalitas serupa serta mengevaluasi kualitas pemetaan menggunakan ukuran stress. Hasil analisis diharapkan memberikan gambaran menyeluruh mengenai distribusi pola kriminalitas di Amerika Serikat serta mendukung pemahaman terhadap hubungan antar variabel kriminalitas dalam data multivariat.

2 Source Code

Berikut merupakan source code beserta penjelasan mengenai coding pada RStudio.

2.1 Library

> library(readxl)
> library(MASS)

readxl digunakan untuk membaca file Excel (.xlsx) ke dalam R. MASS berisi fungsi-fungsi statistik, termasuk cmdscale() untuk MDS klasik.

2.2 Impor Data

> Data <- read_excel("D:/MATERI KULIAH/SMT 5/data_anmul.xlsx")
> head(Data)
# A tibble: 6 × 5
  State      Murder Assault UrbanPop  Rape
  <chr>       <dbl>   <dbl>    <dbl> <dbl>
1 Alabama      13.2     236       58  21.2
2 Alaska       10       263       48  44.5
3 Arizona       8.1     294       80  31  
4 Arkansas      8.8     190       50  19.5
5 California    9       276       91  40.6
6 Colorado      7.9     204       78  38.7
> Data <- Data[,-1]

embaca dataset langsung dari file Excel yang disimpan di komputer pengguna. Setelah data berhasil dimuat, fungsi Data digunakan untuk menampilkan keseluruhan isi dataset, sedangkan head(Data) memberikan enam baris pertama sebagai gambaran awal struktur data. syntax Data <- Data[,-1] digunakan untuk menghapus kolom pertama dari dataset. Kolom ini biasanya berisi nama objek seperti nama negara bagian, yang tidak digunakan dalam perhitungan jarak pada MDS karena bukan variabel numerik.

2.3 Menghitung Jarak Euclidean

> dist_matrix <- as.matrix(dist(Data))

Menghitung jarak Euclidean antar objek menggunakan seluruh variabel numerik dalam dataset. Fungsi dist() menghasilkan objek jarak, lalu as.matrix() mengubahnya menjadi matriks jarak penuh berisi nilai kedekatan antar pasangan objek.

2.4 Mencari Eigen

> A <- dist_matrix^2
> I<-diag(50)
> J<-matrix(rep(1,50), nrow=50, ncol=50)
> V<- I-(1/50)*J
> 
> aa <- V %*% A
> BB <- aa %*% V          
> B <-(-1/2) * BB
> eigen_result <- eigen(B)
> eigenvalues <- eigen_result$values
> eigenvectors <- eigen_result$vectors

Syntax A <- dist_matrix^2 mengubah matriks jarak menjadi jarak kuadrat, yang diperlukan dalam transformasi MDS. Kemudian dibuat matriks identitas I <- diag(50) dan matriks satuan J <- matrix(rep(1,50), nrow=50, ncol=50) sebagai dasar pembentukan centering matrix. Dengan syntax V <- I - (1/50)* J, diperoleh matriks pusat yang berfungsi menghilangkan pengaruh posisi absolut antar objek. Matriks ini kemudian digunakan untuk membentuk matriks B melalui operasi matriks B <- (-1/2) * V %* % A %*% V, yang mencerminkan inner product matrix. Setelah matriks B terbentuk, fungsi eigen(B) digunakan untuk menghitung nilai eigen dan vektor eigen, yang menjadi komponen utama dalam menentukan struktur dimensi rendah.

2.5 Menghitung Tingkat Kumulatif Varians

> cumulative_variance <- cumsum(eigenvalues) / sum(eigenvalues)

Menghitung proporsi variasi yang dijelaskan oleh nilai-nilai eigen secara kumulatif. Pengukuran ini penting untuk menilai seberapa besar informasi dalam data asli yang dapat direpresentasikan dalam satu atau dua dimensi.

2.6 Menentukan Titik Koordinat Objek

> fit <- cmdscale(dist_matrix, k=2)
> # Hitung disparities
> disparities <- matrix(0, nrow = 50, ncol = 50)
> 
> for (i in 1:50) {
+   for (j in 1:50) {
+     disparities[i, j] <- sqrt(sum((fit[i,] - fit[j,])^2))
+   }
+ }

Digunakan untuk langsung mendapatkan koordinat objek dalam dua dimensi. Fungsi ini mengonversi matriks jarak menjadi titik-titik pada bidang dua dimensi dengan mempertahankan struktur jarak sebaik mungkin. Hasilnya disimpan dalam objek fit, yang berisi nilai koordinat untuk setiap objek. Disparities digunakan untuk menghitung ulang jarak antar titik pada ruang dua dimensi menggunakan koordinat MDS. Dengan membuat matriks kosong disparities dan mengisinya melalui loop, setiap pasangan titik dihitung jaraknya menggunakan rumus Euclidean.

2.7 Menghitung Nilai Stress

> stress <- sqrt(sum((dist_matrix - disparities)^2) / sum(dist_matrix^2))
> cat("Nilai Stress:", stress, "\n")
Nilai Stress: 0.01590128

Nilai stress dihitung melalui syntax sqrt(sum((dist_matrix - disparities)^2) / sum(dist_matrix^2)). Stress merupakan ukuran distorsi antara jarak asli dan jarak hasil pemetaan, sehingga semakin kecil nilai stress, semakin baik kualitas pemetaan MDS.

2.8 Membuat Visualisasi Hasil Pemetaan

> Data <- read_excel("D:/MATERI KULIAH/SMT 5/data_anmul.xlsx")
> plot(fit, type="n",
+      xlab="Dimensi 1", ylab="Dimensi 2",
+      main="Pemetaan Amerika Serikat dengan Indikator Kriminalitas")
> 
> points(fit, pch=19, cex=0.8)
> 
> text(fit,
+      labels = Data[[1]],  
+      cex = 0.55,
+      pos = 3,
+      offset = 0.6)

Membuat visualisasi hasil pemetaan menggunakan syntax plot() yang membuat kerangka grafik dua dimensi dengan judul dan label sumbu. Titik koordinat objek ditambahkan menggunakan points(), dan label nama objek ditampilkan melalui fungsi text().

3 Hasil dan Pembahasan

3.1 Deskripsi Data

> Data
# A tibble: 50 × 5
   State       Murder Assault UrbanPop  Rape
   <chr>        <dbl>   <dbl>    <dbl> <dbl>
 1 Alabama       13.2     236       58  21.2
 2 Alaska        10       263       48  44.5
 3 Arizona        8.1     294       80  31  
 4 Arkansas       8.8     190       50  19.5
 5 California     9       276       91  40.6
 6 Colorado       7.9     204       78  38.7
 7 Connecticut    3.3     110       77  11.1
 8 Delaware       5.9     238       72  15.8
 9 Florida       15.4     335       80  31.9
10 Georgia       17.4     211       60  25.8
# ℹ 40 more rows

Output ini menampilkan deskripsi data set yang digunakan untuk analisis. Diketahui data tersebut memiliki 4 variabel yaitu Murder, Assault, UrbanPop, dan Rape.

> Data <- Data[,-1]
> Data
# A tibble: 50 × 4
   Murder Assault UrbanPop  Rape
    <dbl>   <dbl>    <dbl> <dbl>
 1   13.2     236       58  21.2
 2   10       263       48  44.5
 3    8.1     294       80  31  
 4    8.8     190       50  19.5
 5    9       276       91  40.6
 6    7.9     204       78  38.7
 7    3.3     110       77  11.1
 8    5.9     238       72  15.8
 9   15.4     335       80  31.9
10   17.4     211       60  25.8
# ℹ 40 more rows

Output ini menampilkan data yang kolom pertama telah dihapus karena kolom ini berisi nama objek yaitu nama negara bagian yang tidak digunakan dalam perhitungan jarak pada MDS karena bukan variabel numerik sehingga akan memudahkan saat melakukan analisis.

3.2 Menghitung Jarak Matriks

> dist_matrix
           1         2         3         4         5         6          7
1    0.00000  37.17701  63.00833  46.92814  55.52477  41.93256 128.206942
2   37.17701   0.00000  46.59249  77.19741  45.10222  66.47594 159.406556
3   63.00833  46.59249   0.00000 108.85192  23.19418  90.35115 185.159526
4   46.92814  77.19741 108.85192   0.00000  97.58202  36.73486  85.028289
5   55.52477  45.10222  23.19418  97.58202   0.00000  73.19713 169.277110
6   41.93256  66.47594  90.35115  36.73486  73.19713   0.00000  98.081191
7  128.20694 159.40656 185.15953  85.02829 169.27711  98.08119   0.000000
8   16.80625  45.18296  58.61638  53.01038  49.29148  41.47783 128.210179
9  102.00162  79.97450  41.65453 148.73574  60.98073 131.40582 226.303005
10  25.84183  57.03026  86.03796  25.58613  73.99730  25.09303 104.426529
11 191.80305 221.19354 248.26897 147.77598 231.07109 159.17918  64.952367
12 116.76198 146.48498 176.81767  70.58704 162.61279  90.88641  25.280427
13  28.45488  42.91165  45.69781  67.77027  32.71880  47.66907 139.906469
14 123.34521 152.80409 181.89780  78.47809 166.22996  93.61506  16.316250
15 180.61010 209.98352 239.99146 134.59495 224.63466 152.07975  57.595573
16 121.51987 151.48020 180.02891  76.75344 164.51675  92.17972  14.173920
17 127.28417 156.61204 187.69030  81.09285 173.20791 101.02475  26.343880
18  15.45445  32.34888  48.49464  61.54551  41.63556  49.97499 140.398077
19 154.14529 183.89753 214.32741 107.85073 199.93111 127.90016  37.647443
20  64.99362  44.83949  15.01599 111.64291  36.34735  97.30041 191.161947
21  91.64851 123.25421 145.87591  54.18118 129.52471  59.90000  40.165284
22  28.48543  28.85775  39.87242  71.10028  27.74635  51.45483 147.266561
23 164.65096 194.25357 223.08826 119.32464 207.22254 134.76454  39.746698
24  27.39014  28.63512  52.70873  69.68536  55.68357  68.66440 153.263955
25  59.78829  89.30672 116.46738  24.89438 100.98891  29.17979  70.695827
26 127.39262 156.67358 187.54085  81.16311 172.99607 100.75167  24.746313
27 134.43697 164.11426 193.42360  88.97893 178.10081 105.66835  17.865050
28  37.43047  34.88682  44.79743  74.28869  26.74696  48.83421 146.551083
29 179.73620 209.25441 239.25562 133.67831 224.05539 151.58918  57.043843
30  83.24302 114.73557 135.85040  49.84426 119.04117  50.42083  51.196680
31  51.64349  33.52193  13.89604  97.93120  24.49510  81.73622 176.580322
32  33.71083  43.18298  40.85352  73.76212  26.90093  52.27810 145.268166
33 101.96102  79.37607  57.61961 147.18424  80.33212 138.97759 229.504009
34 192.41614 221.37859 252.80819 145.85554 238.21446 165.75093  73.038962
35 117.38761 147.37334 174.33818  74.36975 157.99851  85.81754  15.036289
36  85.84870 116.42942 143.93141  43.01267 128.77935  57.09974  43.036031
37  78.38686 106.93012 135.67288  36.89512 120.03958  47.36412  53.242840
38 131.08509 161.60090 188.86622  86.99086 172.99936 101.03960   8.027453
39  70.33811 103.90380 122.41887  42.18531 107.21311  43.87949  64.837104
40  44.18292  27.55649  36.89092  89.24887  47.06134  82.64194 172.206765
41 151.08911 179.94813 211.75158 104.45521 197.52438 125.30211  40.039231
42  48.34760  77.88453 108.25812  12.61428  94.72766  28.00589  82.192761
43  41.56609  72.36221  93.27599  32.74462  77.38023  14.50103  92.659160
44 118.50270 148.27609 174.25734  76.43900 157.49263  85.62552  15.755951
45 190.37069 218.29047 251.48926 143.52857 237.43546 165.04769  76.617361
46  80.29533 110.64669 139.42471  36.42156 124.82091  53.41685  49.307200
47  92.82047 122.14700 149.29786  51.20478 133.10657  60.64206  38.334058
48 156.79241 185.64086 218.00608 110.07111 204.25371 132.36011  47.895720
49 183.77573 213.57540 242.31238 138.34424 226.45750 154.11522  58.056696
50  75.50709 106.74010 135.38039  30.98726 121.72034  52.03672  54.060152
           8         9        10        11        12         13         14
1   16.80625 102.00162  25.84183 191.80305 116.76198  28.454877 123.345207
2   45.18296  79.97450  57.03026 221.19354 146.48498  42.911653 152.804090
3   58.61638  41.65453  86.03796 248.26897 176.81767  45.697812 181.897801
4   53.01038 148.73574  25.58613 147.77598  70.58704  67.770274  78.478086
5   49.29148  60.98073  73.99730 231.07109 162.61279  32.718802 166.229961
6   41.47783 131.40582  25.09303 159.17918  90.88641  47.669068  93.615063
7  128.21018 226.30300 104.42653  64.95237  25.28043 139.906469  16.316250
8    0.00000  99.10832  33.24530 192.36611 119.42131  18.151859 125.310534
9   99.10832   0.00000 125.76649 289.42857 217.66518  86.558708 222.923866
10  33.24530 125.76649   0.00000 167.12800  93.11606  45.002667  98.772871
11 192.36611 289.42857 167.12800   0.00000  79.75143 203.099606  69.406412
12 119.42131 217.66518  93.11606  79.75143   0.00000 132.811445  15.407790
13  18.15186  86.55871  45.00267 203.09961 132.81145   0.000000 137.256111
14 125.31053 222.92387  98.77287  69.40641  15.40779 137.256111   0.000000
15 182.70999 281.01352 156.44581  29.40782  64.13712 195.329286  58.584042
16 123.16594 221.08272  97.17407  71.10084  13.96424 135.278823   3.929377
17 130.59743 228.33276 103.04145  70.45970  13.40970 143.599373  14.606163
18  16.97675  87.67035  38.69057 203.97061 130.43328  17.811232 136.255936
19 156.66665 255.15231 130.47256  50.56679  37.67240 170.033320  36.003472
20  63.57798  37.78386  89.50536 254.68757 181.18954  53.593376 187.179192
21  89.95832 187.04374  68.76227 103.09714  42.53998 100.495224  41.544314
22  26.53168  80.35627  47.39810 209.83386 138.39097  15.591664 143.065789
23 166.14166 264.22583 140.32783  31.62040  49.48232 178.213636  41.706834
24  36.47917  85.39046  51.35543 216.83231 140.04164  41.244394 147.822258
25  61.37891 157.49175  35.57134 132.93115  62.10443  72.315973  65.613108
26 130.39314 228.32786 103.30208  69.88512  11.76435 143.447273  13.512957
27 136.37833 234.46401 110.19573  59.93071  19.90427 148.806787  12.596031
28  35.05325  84.25586  50.56758 207.73360 138.76743  22.366046 142.221658
29 181.85469 280.24748 155.66560  31.22067  63.20870 194.607657  58.096988
30  80.87799 176.89717  60.77829 113.18732  52.82234  90.399336  51.931493
31  50.08932  51.14724  75.17772 239.72655 166.96961  39.135789 172.481448
32  24.18946  81.54220  50.64366 208.18609 138.54191   6.236986 142.699755
33 102.86156  38.52791 127.33597 293.60024 217.44372  96.214188 225.019221
34 195.27227 293.62275 168.61142  41.33594  75.99901 208.585834  72.757474
35 118.17919 215.46661  92.88364  74.46771  22.69207 129.311136  12.213517
36  87.19593 184.98392  61.75986 106.07417  34.73672  99.294713  38.136072
37  80.30722 176.81066  54.05090 114.49004  43.85544  91.729712  46.842075
38 132.00367 229.94958 106.82238  61.23798  23.11233 143.769329  11.662761
39  66.20801 163.31246  50.99265 128.62822  63.56453  77.048621  66.187083
40  48.72515  65.18712  69.19458 235.80098 159.76586  46.295248 167.030207
41 154.42283 252.43884 127.29478  55.68671  35.21931 167.874953  34.753417
42  53.34323 148.59287  23.42755 144.38594  70.16160  65.675338  75.708718
43  39.66522 134.17992  22.85126 155.29601  86.41007  48.171984  89.551661
44 118.51456 215.53385  94.29236  74.13973  27.42353 129.240280  17.135052
45 194.25460 292.02008 166.72492  51.91926  75.34693 207.925660  73.722724
46  82.67872 180.28602  56.02874 111.85030  38.13214  95.202416  43.067157
47  93.60433 190.55563  68.59096  99.69298  33.64461 104.698615  33.545193
48 160.56242 258.46054 133.22965  57.27102  42.18554 174.350738  42.885196
49 185.19420 283.42380 159.51188  20.82426  68.15101 197.332410  61.042608
50  77.93491 176.11261  52.11909 117.37721  41.67253  91.414003  48.562537
           15         16         17        18         19        20        21
1  180.610105 121.519875 127.284170  15.45445 154.145289  64.99362  91.64851
2  209.983523 151.480197 156.612037  32.34888 183.897526  44.83949 123.25421
3  239.991458 180.028914 187.690303  48.49464 214.327413  15.01599 145.87591
4  134.594948  76.753436  81.092848  61.54551 107.850730 111.64291  54.18118
5  224.634659 164.516747 173.207910  41.63556 199.931113  36.34735 129.52471
6  152.079749  92.179716 101.024749  49.97499 127.900156  97.30041  59.90000
7   57.595573  14.173920  26.343880 140.39808  37.647443 191.16195  40.16528
8  182.709989 123.165945 130.597435  16.97675 156.666652  63.57798  89.95832
9  281.013523 221.082722 228.332762  87.67035 255.152307  37.78386 187.04374
10 156.445805  97.174071 103.041448  38.69057 130.472564  89.50536  68.76227
11  29.407822  71.100844  70.459705 203.97061  50.566788 254.68757 103.09714
12  64.137119  13.964240  13.409698 130.43328  37.672404 181.18954  42.53998
13 195.329286 135.278823 143.599373  17.81123 170.033320  53.59338 100.49522
14  58.584042   3.929377  14.606163 136.25594  36.003472 187.17919  41.54431
15   0.000000  60.177487  53.993055 193.96662  27.879383 244.93072  97.27713
16  60.177487   0.000000  15.766420 134.39494  36.989863 185.33788  39.01859
17  53.993055  15.766420   0.000000 140.93722  28.407921 191.93960  52.12571
18 193.966621 134.394940 140.937220   0.00000 167.825058  51.47980 102.55150
19  27.879383  36.989863  28.407921 167.82506   0.000000 218.69989  74.76323
20 244.930725 185.337881 191.939600  51.47980 218.699886   0.00000 152.65929
21  97.277130  39.018585  52.125713 102.55150  74.763226 152.65929   0.00000
22 201.381355 141.398798 148.859665  16.65233 175.949680  46.12949 108.48839
23  18.713899  43.237715  40.199005 177.60512  19.919086 228.52871  79.34009
24 203.972670 146.023354 150.351588  24.70830 176.939227  48.45132 117.97682
25 124.035680  64.015936  72.298686  71.65166  99.246007 122.05921  35.05382
26  53.529898  14.406943   3.834058 141.03546  27.733914 191.92459  51.25007
27  46.609548  13.789126  13.349157 147.58286  23.717715 198.50866  52.32638
28 200.738860 140.770878 148.923940  28.47244 176.131343  53.21701 107.55431
29   2.291288  59.594127  53.141321 193.13772  26.530925 244.10967  96.72916
30 108.241813  49.674943  62.293980  93.29823  85.843404 143.04269  11.45644
31 230.493557 170.716051 177.630318  37.76255 204.553123  15.89025 137.91171
32 200.856292 140.757309 149.261515  21.41728 175.732439  49.79890 105.67370
33 281.504316 223.108964 228.131388  90.70816 254.439973  44.64056 192.40062
34  17.548789  74.333909  65.728304 206.24056  38.664454 257.06906 112.20945
35  67.439009  10.920165  26.110726 129.56948  46.443514 180.33569  31.23171
36  96.130380  36.115924  45.202876  98.43922  71.288779 149.28138  17.65021
37 105.072784  45.452173  54.009629  90.89367  80.635538 141.15314  24.35672
38  52.485903  11.256109  20.555291 143.60049  32.218783 194.55696  44.98411
39 121.796716  63.425941  74.523084  80.02006  97.872059 129.30066  26.34388
40 223.795174 165.259826 170.224939  35.00129 196.957813  28.97758 135.67402
41  32.385336  36.247483  25.001200 165.02400   8.537564 215.78056  74.71017
42 133.388005  74.222975  80.091260  61.61923 107.596561 112.30503  48.85489
43 147.871194  87.710547  96.652160  50.18147 123.250355  99.88619  53.68920
44  68.996812  15.901258  31.477135 130.33162  49.388460 180.71696  30.18278
45  26.249000  75.535952  64.832554 204.57820  39.969613 255.12226 114.19654
46 100.816913  41.273963  48.485049  93.31592  75.368229 144.25758  23.85728
47  91.663788  31.941196  43.214581 104.93312  68.338642 155.29601  16.06767
48  31.068473  44.280696  31.906112 170.91957  12.775367 221.62719  82.40564
49   9.508417  62.509199  58.417977 196.74806  33.678628 247.73916  98.03311
50 105.231412  46.458584  52.696300  88.86799  79.043849 139.78230  27.84331
          22        23        24        25         26        27        28
1   28.48543 164.65096  27.39014  59.78829 127.392621 134.43697  37.43047
2   28.85775 194.25357  28.63512  89.30672 156.673578 164.11426  34.88682
3   39.87242 223.08826  52.70873 116.46738 187.540849 193.42360  44.79743
4   71.10028 119.32464  69.68536  24.89438  81.163107  88.97893  74.28869
5   27.74635 207.22254  55.68357 100.98891 172.996069 178.10081  26.74696
6   51.45483 134.76454  68.66440  29.17979 100.751675 105.66835  48.83421
7  147.26656  39.74670 153.26396  70.69583  24.746313  17.86505 146.55108
8   26.53168 166.14166  36.47917  61.37891 130.393136 136.37833  35.05325
9   80.35627 264.22583  85.39046 157.49175 228.327856 234.46401  84.25586
10  47.39810 140.32783  51.35543  35.57134 103.302081 110.19573  50.56758
11 209.83386  31.62040 216.83231 132.93115  69.885120  59.93071 207.73360
12 138.39097  49.48232 140.04164  62.10443  11.764353  19.90427 138.76743
13  15.59166 178.21364  41.24439  72.31597 143.447273 148.80679  22.36605
14 143.06579  41.70683 147.82226  65.61311  13.512957  12.59603 142.22166
15 201.38135  18.71390 203.97267 124.03568  53.529898  46.60955 200.73886
16 141.39880  43.23772 146.02335  64.01594  14.406943  13.78913 140.77088
17 148.85967  40.19900 150.35159  72.29869   3.834058  13.34916 148.92394
18  16.65233 177.60512  24.70830  71.65166 141.035457 147.58286  28.47244
19 175.94968  19.91909 176.93923  99.24601  27.733914  23.71771 176.13134
20  46.12949 228.52871  48.45132 122.05921 191.924595 198.50866  53.21701
21 108.48839  79.34009 117.97682  35.05382  51.250073  52.32638 107.55431
22   0.00000 184.52480  35.44009  77.47400 148.808266 154.78953  13.29737
23 184.52480   0.00000 188.77871 107.09146  39.384515  30.34996 183.52782
24  35.44009 188.77871   0.00000  86.08496 150.610425 158.46956  47.62793
25  77.47400 107.09146  86.08496   0.00000  72.098821  77.45308  76.96805
26 148.80827  39.38451 150.61042  72.09882   0.000000  11.52823 148.82070
27 154.78953  30.34996 158.46956  77.45308  11.528226   0.00000 154.25194
28  13.29737 183.52782  47.62793  76.96805 148.820697 154.25194   0.00000
29 200.70715  18.82870 202.98217 123.42731  52.686051  45.98739 200.21054
30  98.63458  90.19590 110.01627  28.51175  61.674306  63.18940  97.34413
31  30.42187 213.90776  39.98862 107.09795 177.596875 183.97492  37.46799
32  15.06652 183.63006  43.53160  77.72271 149.111032 154.33211  20.64510
33  89.03263 266.03295  78.07439 161.45715 228.247870 235.77500  97.03427
34 214.24409  35.69832 214.76995 137.36466  65.474041  60.57962 214.11597
35 135.78192  49.48141 142.75129  58.63557  25.133444  22.93491 134.49599
36 105.40522  79.28569 111.07952  28.39014  44.747290  49.54331 105.24904
37  96.69788  88.21678 103.93883  19.69822  53.512802  58.63617  95.79760
38 150.48814  34.71253 155.86494  73.29516  19.296114  11.07068 149.66302
39  86.74059 104.35061  96.50249  27.06234  73.846936  76.66192  87.28534
40  37.63044 208.24901  21.16719 103.66605 170.390053 177.94103  48.73490
41 173.11320  25.34995 173.49288  96.71194  24.714368  23.64191 173.27666
42  69.15526 117.29983  73.28335  15.50258  80.243941  87.13421  70.32645
43  55.17717 130.57320  68.86305  25.49471  96.543772 101.37337  54.97727
44 135.97445  50.63842 144.28115  59.37786  30.001500  26.27109 134.31162
45 212.79619  41.78445 211.87973 136.67202  64.834250  62.03628 212.90383
46 100.70909  84.45283 105.07483  24.27962  48.308798  54.33489 100.96425
47 110.66083  74.21172 118.60110  33.57082  42.379712  45.43325 109.41791
48 179.46476  29.16093 178.54411 103.62480  32.101402  31.99687 179.98372
49 203.83508  19.43759 207.70638 126.43069  57.861213  49.52151 202.87237
50  97.16141  89.29894  99.74337  23.50745  52.481235  59.09365  98.36320
           29        30        31         32        33        34         35
1  179.736196  83.24302  51.64349  33.710829 101.96102 192.41614 117.387606
2  209.254415 114.73557  33.52193  43.182983  79.37607 221.37859 147.373335
3  239.255616 135.85040  13.89604  40.853519  57.61961 252.80819 174.338177
4  133.678308  49.84426  97.93120  73.762118 147.18424 145.85554  74.369752
5  224.055395 119.04117  24.49510  26.900929  80.33212 238.21446 157.998513
6  151.589182  50.42083  81.73622  52.278102 138.97759 165.75093  85.817539
7   57.043843  51.19668 176.58032 145.268166 229.50401  73.03896  15.036289
8  181.854695  80.87799  50.08932  24.189461 102.86156 195.27227 118.179186
9  280.247480 176.89717  51.14724  81.542198  38.52791 293.62275 215.466610
10 155.665603  60.77829  75.17772  50.643657 127.33597 168.61142  92.883637
11  31.220666 113.18732 239.72655 208.186095 293.60024  41.33594  74.467711
12  63.208702  52.82234 166.96961 138.541907 217.44372  75.99901  22.692069
13 194.607657  90.39934  39.13579   6.236986  96.21419 208.58583 129.311136
14  58.096988  51.93149 172.48145 142.699755 225.01922  72.75747  12.213517
15   2.291288 108.24181 230.49356 200.856292 281.50432  17.54879  67.439009
16  59.594127  49.67494 170.71605 140.757309 223.10896  74.33391  10.920165
17  53.141321  62.29398 177.63032 149.261515 228.13139  65.72830  26.110726
18 193.137723  93.29823  37.76255  21.417283  90.70816 206.24056 129.569479
19  26.530925  85.84340 204.55312 175.732439 254.43997  38.66445  46.443514
20 244.109668 143.04269  15.89025  49.798896  44.64056 257.06906 180.335687
21  96.729158  11.45644 137.91171 105.673696 192.40062 112.20945  31.231715
22 200.707150  98.63458  30.42187  15.066519  89.03263 214.24409 135.781921
23  18.828701  90.19590 213.90776 183.630063 266.03295  35.69832  49.481411
24 202.982167 110.01627  39.98862  43.531598  78.07439 214.76995 142.751287
25 123.427307  28.51175 107.09795  77.722712 161.45715 137.36466  58.635569
26  52.686051  61.67431 177.59687 149.111032 228.24787  65.47404  25.133444
27  45.987390  63.18940 183.97492 154.332109 235.77500  60.57962  22.934908
28 200.210539  97.34413  37.46799  20.645096  97.03427 214.11597 134.495985
29   0.000000 107.73848 229.73300 200.163833 280.50556  17.16188  67.071976
30 107.738480   0.00000 128.17913  95.399057 183.46294 123.27534  41.518309
31 229.732997 128.17913   0.00000  35.399011  59.89624 242.90615 165.472959
32 200.163833  95.39906  35.39901   0.000000  93.13222 214.25342 134.586515
33 280.505561 183.46294  59.89624  93.132218   0.00000 292.38892 219.202144
34  17.161876 123.27534 242.90615 214.253425 292.38892   0.00000  82.626025
35  67.071976  41.51831 165.47296 134.586515 219.20214  82.62602   0.000000
36  95.448939  22.51844 134.64565 104.835395 187.56644 109.57614  31.819019
37 104.522151  24.50510 126.23427  97.132281 180.02180 118.43061  40.659193
38  51.998077  55.80699 179.90789 149.159244 232.67215  67.77175  15.755951
39 121.050114  18.84808 115.06711  82.323326 168.77796 136.00647  56.980874
40 222.863837 127.05782  24.92308  45.747678  58.44587 234.92126 161.436117
41  31.237477  85.52169 201.63395 173.594873 251.19023  41.48795  46.284015
42 132.649802  42.89814  97.77668  71.344586 150.04559 145.64313  70.321121
43 147.242521  43.79475  84.86018  53.365907 140.74605 161.56995  81.436908
44  68.744236  40.45306 165.76152 134.404799 220.12787  84.67656   6.637771
45  25.688519 125.11914 241.11087 213.666141 289.53523  13.04492  84.634804
46 100.081966  26.26442 129.72421 100.841063 182.00662 113.66992  37.972753
47  91.189363  22.76664 140.35159 110.001909 194.50195 105.86997  25.747427
48  29.631065  93.15117 207.67602 180.137170 256.26457  36.72887  54.460720
49  10.860018 108.86707 233.17618 202.749451 285.01447  23.73794  68.588993
50 104.361391  29.25064 125.57631  97.230345 176.74753 117.54527  44.044182
          36        37         38        39        40         41        42
1   85.84870  78.38686 131.085087  70.33811  44.18292 151.089113  48.34760
2  116.42942 106.93012 161.600897 103.90380  27.55649 179.948131  77.88453
3  143.93141 135.67288 188.866222 122.41887  36.89092 211.751576 108.25812
4   43.01267  36.89512  86.990862  42.18531  89.24887 104.455206  12.61428
5  128.77935 120.03958 172.999364 107.21311  47.06134 197.524378  94.72766
6   57.09974  47.36412 101.039596  43.87949  82.64194 125.302115  28.00589
7   43.03603  53.24284   8.027453  64.83710 172.20677  40.039231  82.19276
8   87.19593  80.30722 132.003674  66.20801  48.72515 154.422829  53.34323
9  184.98392 176.81066 229.949581 163.31246  65.18712 252.438844 148.59287
10  61.75986  54.05090 106.822376  50.99265  69.19458 127.294776  23.42755
11 106.07417 114.49004  61.237978 128.62822 235.80098  55.686713 144.38594
12  34.73672  43.85544  23.112334  63.56453 159.76586  35.219313  70.16160
13  99.29471  91.72971 143.769329  77.04862  46.29525 167.874953  65.67534
14  38.13607  46.84208  11.662761  66.18708 167.03021  34.753417  75.70872
15  96.13038 105.07278  52.485903 121.79672 223.79517  32.385336 133.38801
16  36.11592  45.45217  11.256109  63.42594 165.25983  36.247483  74.22297
17  45.20288  54.00963  20.555291  74.52308 170.22494  25.001200  80.09126
18  98.43922  90.89367 143.600487  80.02006  35.00129 165.023998  61.61923
19  71.28878  80.63554  32.218783  97.87206 196.95781   8.537564 107.59656
20 149.28138 141.15314 194.556958 129.30066  28.97758 215.780560 112.30503
21  17.65021  24.35672  44.984108  26.34388 135.67402  74.710173  48.85489
22 105.40522  96.69788 150.488139  86.74059  37.63044 173.113200  69.15526
23  79.28569  88.21678  34.712534 104.35061 208.24901  25.349951 117.29983
24 111.07952 103.93883 155.864942  96.50249  21.16719 173.492882  73.28335
25  28.39014  19.69822  73.295157  27.06234 103.66605  96.711943  15.50258
26  44.74729  53.51280  19.296114  73.84694 170.39005  24.714368  80.24394
27  49.54331  58.63617  11.070682  76.66192 177.94103  23.641912  87.13421
28 105.24904  95.79760 149.663021  87.28534  48.73490 173.276657  70.32645
29  95.44894 104.52215  51.998077 121.05011 222.86384  31.237477 132.64980
30  22.51844  24.50510  55.806989  18.84808 127.05782  85.521693  42.89814
31 134.64565 126.23427 179.907893 115.06711  24.92308 201.633951  97.77668
32 104.83539  97.13228 149.159244  82.32333  45.74768 173.594873  71.34459
33 187.56644 180.02180 232.672151 168.77796  58.44587 251.190227 150.04559
34 109.57614 118.43061  67.771749 136.00647 234.92126  41.487950 145.64313
35  31.81902  40.65919  15.755951  56.98087 161.43612  46.284015  70.32112
36   0.00000  12.42497  45.465371  32.20450 129.81175  69.380689  39.25774
37  12.42497   0.00000  55.166294  32.68409 122.05527  78.015768  31.29936
38  45.46537  55.16629   0.000000  70.00693 175.00963  33.758851  84.17013
39  32.20450  32.68409  70.006928   0.00000 113.44003  97.613575  37.70942
40 129.81175 122.05527 175.009628 113.44003   0.00000 193.557356  91.77581
41  69.38069  78.01577  33.758851  97.61357 193.55736   0.000000 104.34160
42  39.25774  31.29936  84.170125  37.70942  91.77581 104.341602   0.00000
43  52.07168  44.81384  96.136986  34.06362  84.37944 121.204373  24.74288
44  33.54057  41.63952  18.264994  56.37553 162.57491  49.833423  71.97916
45 109.55273 117.81723  70.671777 137.51673 232.14980  40.225862 143.86275
46   7.35527  10.57922  51.180856  32.85985 124.06470  72.859454  33.17424
47  10.30534  15.56984  40.681445  36.84793 136.75836  66.667833  46.15333
48  76.52647  85.25632  41.781814 104.68663 198.83393   8.766984 110.52154
49  98.53243 107.63150  53.623689 122.83684 227.32384  39.184691 136.54878
50  13.54253  15.63010  56.300444  31.02982 119.05196  76.595300  29.97749
          43         44        45         46        47         48         49
1   41.56609 118.502700 190.37069  80.295330  92.82047 156.792411 183.775733
2   72.36221 148.276094 218.29047 110.646690 122.14700 185.640863 213.575397
3   93.27599 174.257338 251.48926 139.424711 149.29786 218.006078 242.312381
4   32.74462  76.438995 143.52857  36.421560  51.20478 110.071113 138.344245
5   77.38023 157.492635 237.43546 124.820912 133.10657 204.253715 226.457502
6   14.50103  85.625522 165.04769  53.416851  60.64206 132.360115 154.115217
7   92.65916  15.755951  76.61736  49.307200  38.33406  47.895720  58.056696
8   39.66522 118.514556 194.25460  82.678716  93.60433 160.562418 185.194195
9  134.17992 215.533849 292.02008 180.286023 190.55563 258.460539 283.423799
10  22.85126  94.292364 166.72492  56.028743  68.59096 133.229651 159.511880
11 155.29601  74.139733  51.91926 111.850302  99.69298  57.271022  20.824265
12  86.41007  27.423530  75.34693  38.132139  33.64461  42.185543  68.151009
13  48.17198 129.240280 207.92566  95.202416 104.69862 174.350738 197.332410
14  89.55166  17.135052  73.72272  43.067157  33.54519  42.885196  61.042608
15 147.87119  68.996812  26.24900 100.816913  91.66379  31.068473   9.508417
16  87.71055  15.901258  75.53595  41.273963  31.94120  44.280696  62.509199
17  96.65216  31.477135  64.83255  48.485049  43.21458  31.906112  58.417977
18  50.18147 130.331616 204.57820  93.315915 104.93312 170.919572 196.748062
19 123.25035  49.388460  39.96961  75.368229  68.33864  12.775367  33.678628
20  99.88619 180.716961 255.12226 144.257582 155.29601 221.627187 247.739157
21  53.68920  30.182777 114.19654  23.857284  16.06767  82.405643  98.033107
22  55.17717 135.974446 212.79619 100.709086 110.66083 179.464760 203.835080
23 130.57320  50.638424  41.78445  84.452827  74.21172  29.160933  19.437592
24  68.86305 144.281149 211.87973 105.074830 118.60110 178.544112 207.706379
25  25.49471  59.377858 136.67202  24.279621  33.57082 103.624804 126.430692
26  96.54377  30.001500  64.83425  48.308798  42.37971  32.101402  57.861213
27 101.37337  26.271087  62.03628  54.334888  45.43325  31.996875  49.521510
28  54.97727 134.311615 212.90383 100.964251 109.41791 179.983721 202.872374
29 147.24252  68.744236  25.68852 100.081966  91.18936  29.631065  10.860018
30  43.79475  40.453059 125.11914  26.264425  22.76664  93.151167 108.867075
31  84.86018 165.761515 241.11087 129.724207 140.35159 207.676022 233.176178
32  53.36591 134.404799 213.66614 100.841063 110.00191 180.137170 202.749451
33 140.74605 220.127872 289.53523 182.006621 194.50195 256.264570 285.014473
34 161.56995  84.676561  13.04492 113.669917 105.86997  36.728871  23.737944
35  81.43691   6.637771  84.63480  37.972753  25.74743  54.460720  68.588993
36  52.07168  33.540572 109.55273   7.355270  10.30534  76.526466  98.532431
37  44.81384  41.639524 117.81723  10.579225  15.56984  85.256319 107.631501
38  96.13699  18.264994  70.67178  51.180856  40.68145  41.781814  53.623689
39  34.06362  56.375527 137.51673  32.859854  36.84793 104.686628 122.836843
40  84.37944 162.574906 232.14980 124.064701 136.75836 198.833926 227.323844
41 121.20437  49.833423  40.22586  72.859454  66.66783   8.766984  39.184691
42  24.74288  71.979164 143.86275  33.174237  46.15333 110.521536 136.548782
43   0.00000  81.596630 161.33115  48.525045  57.10674 128.032965 149.726751
44  81.59663   0.000000  87.32634  40.223501  26.18263  58.251266  69.510934
45 161.33115  87.326342   0.00000 112.937770 106.38722  33.968515  34.370336
46  48.52505  40.223501 112.93777   0.000000  16.47726  79.616581 103.686161
47  57.10674  26.182628 106.38722  16.477257   0.00000  74.434535  93.552766
48 128.03296  58.251266  33.96851  79.616581  74.43453   0.000000  39.049456
49 149.72675  69.510934  34.37034 103.686161  93.55277  39.049456   0.000000
50  46.18246  46.338429 116.59125   7.930952  23.34952  82.957218 108.354418
           50
1   75.507086
2  106.740105
3  135.380390
4   30.987255
5  121.720335
6   52.036718
7   54.060152
8   77.934909
9  176.112606
10  52.119094
11 117.377212
12  41.672533
13  91.414003
14  48.562537
15 105.231412
16  46.458584
17  52.696300
18  88.867992
19  79.043849
20 139.782295
21  27.843312
22  97.161412
23  89.298936
24  99.743371
25  23.507446
26  52.481235
27  59.093654
28  98.363205
29 104.361391
30  29.250641
31 125.576311
32  97.230345
33 176.747532
34 117.545268
35  44.044182
36  13.542526
37  15.630099
38  56.300444
39  31.029824
40 119.051963
41  76.595300
42  29.977492
43  46.182464
44  46.338429
45 116.591252
46   7.930952
47  23.349518
48  82.957218
49 108.354418
50   0.000000

Output ini menampilkan Matriks jarak di atas merupakan hasil perhitungan jarak Euclidean antar objek (misalnya provinsi, negara bagian, atau unit observasi lain) berdasarkan variabel numerik dalam dataset. Setiap nilai menggambarkan tingkat ketidakmiripan antara dua objek: semakin besar angkanya, semakin tidak mirip objek tersebut; semakin kecil angkanya, semakin mirip atau dekat.

1. Diagonal bernilai 0

Pada baris 1 kolom 1, baris 2 kolom 2, dan seterusnya, terlihat nilai 0.00000. Ini berarti jarak suatu objek terhadap dirinya sendiri adalah nol. Ini adalah karakteristik wajar pada matriks jarak.

2. Pola kedekatan antar objek

Jika melihat beberapa nilai kecil, misalnya:

  • Baris 1 kolom 7 → 16.80625
  • Baris 1 kolom 9 → 102.00162
  • Baris 2 kolom 3 → 46.59249
  • Baris 4 kolom 7 → 23.11023
  • Baris 5 kolom 8 → 18.029110

Nilai-nilai ini menunjukkan bahwa objek-objek tersebut lebih mirip dibandingkan dengan pasangan lainnya yang memiliki jarak lebih besar. Misalnya:

  • Objek 1 dan 7 memiliki jarak 16.80, menunjukkan kedekatan yang cukup tinggi.
  • Objek 5 dan 8 juga memiliki jarak kecil (18.02), sehingga keduanya bisa saja berada dalam cluster yang sama pada hasil MDS.

3. Pasangan objek yang sangat jauh

Beberapa nilai sangat besar, misalnya:

  • Baris 3 kolom 4 → 108.85190
  • Baris 10 kolom 16 → 289.42857
  • Baris 11 kolom 20 → 254.68757

Hal ini mengindikasikan bahwa objek-objek tersebut sangat berbeda berdasarkan variabel-variabel dalam dataset. Dalam peta MDS nanti, pasangan ini akan berada jauh terpisah.

4. Struktur data secara umum

Jika diperhatikan, angka-angka dalam matriks tidak berkelompok di kisaran tertentu. Ada:

  • Jarak kecil (< 30)
  • Jarak sedang (40–100)
  • Jarak besar (> 150)
  • Jarak sangat besar (> 250)

Hal ini menunjukkan heterogenitas yang cukup tinggi dalam dataset. Artinya, objek-objek tidak memiliki pola kemiripan yang homogen; ada objek yang sangat mirip, tetapi ada juga yang sangat berbeda.

3.3 Mencari Nilai Eigen

> eigen_result
eigen() decomposition
$values
 [1]  3.435446e+05  9.897626e+03  2.063520e+03  3.020481e+02  1.524441e-10
 [6]  3.530654e-11  3.485905e-11  3.108343e-11  1.402092e-11  1.174180e-11
[11]  1.146043e-11  1.124426e-11  1.064627e-11  1.042458e-11  8.667796e-12
[16]  8.585363e-12  7.708470e-12  7.333934e-12  4.126079e-12  3.773216e-12
[21]  3.726278e-12  3.519283e-12  2.779982e-12  1.954720e-12  1.898374e-12
[26]  1.017076e-12  9.584985e-14 -1.415291e-13 -3.462516e-13 -3.835016e-13
[31] -8.003654e-13 -1.111324e-12 -1.465002e-12 -2.318104e-12 -2.843200e-12
[36] -3.872565e-12 -4.438350e-12 -4.766850e-12 -7.159163e-12 -7.583835e-12
[41] -8.245047e-12 -8.504123e-12 -1.024295e-11 -1.082686e-11 -1.515440e-11
[46] -1.600053e-11 -1.666295e-11 -2.288197e-11 -3.678044e-11 -8.912223e-11

$vectors
              [,1]         [,2]          [,3]        [,4]         [,5]
 [1,] -0.110559974  0.115070601  0.0549230353  0.13854810  0.000000000
 [2,] -0.158374348  0.180757049 -0.4430630613 -0.23556719 -0.330907665
 [3,] -0.211674702 -0.088759533  0.0371472066 -0.25050650 -0.065395615
 [4,] -0.031290218  0.167900759 -0.0046270736 -0.02997743  0.250852680
 [5,] -0.183275961 -0.226362360 -0.1485025226 -0.16178952 -0.063627475
 [6,] -0.059673070 -0.137903543 -0.2703158440 -0.09905122  0.302852490
 [7,]  0.103880733 -0.129992406  0.1853709424 -0.04027165 -0.115241162
 [8,] -0.113850834 -0.013607811  0.2483371129 -0.21450414  0.276430304
 [9,] -0.281925975 -0.063070571  0.0659960001  0.07179024 -0.024503114
[10,] -0.069157692  0.073278451 -0.0794595792  0.42254726  0.039407188
[11,]  0.210766860 -0.244165103 -0.0819892633  0.19982413  0.099373138
[12,]  0.088371667  0.095180363  0.0334624255 -0.19263054 -0.113535572
[13,] -0.134769639 -0.129635879  0.1295132090  0.02115366 -0.046820886
[14,]  0.098188584 -0.028609467 -0.0822912413  0.09490649 -0.160001872
[15,]  0.197204409  0.033593704  0.0143976931 -0.05002989 -0.228323685
[16,]  0.095183660 -0.031735200 -0.0084613286  0.03756095 -0.072954325
[17,]  0.106432909  0.107283279 -0.0492469047  0.22303344 -0.025488635
[18,] -0.133550677  0.043170722  0.0842660839  0.25797930 -0.099561753
[19,]  0.152289648  0.115470853  0.1032978343 -0.12176391 -0.133072932
[20,] -0.220652144  0.050328595  0.0516702708 -0.11095376  0.047460888
[21,]  0.036282734 -0.195505097  0.1652611113 -0.05954240 -0.094311089
[22,] -0.145790171 -0.059350144 -0.1423049485  0.02871469 -0.035259238
[23,]  0.168828341 -0.052364735 -0.0001447137 -0.04211251 -0.002570131
[24,] -0.148186975  0.275699048  0.1101448106  0.22323720 -0.184417297
[25,] -0.013625531 -0.053028550 -0.1210888443  0.03909228  0.010573420
[26,]  0.106604296  0.095595605 -0.0404692482  0.01415128 -0.109328787
[27,]  0.117886683  0.002122853 -0.0103029331 -0.03777815  0.051641114
[28,] -0.142654419 -0.151800861 -0.3497710768  0.01922930 -0.069975479
[29,]  0.195823419  0.047589808  0.0502440852 -0.05385132 -0.257498243
[30,]  0.018452876 -0.232566892  0.1389107520  0.09277738 -0.050505200
[31,] -0.195978348  0.003381886 -0.0497792196 -0.07947555  0.097639318
[32,] -0.143815690 -0.160061068  0.1039330211  0.05132586 -0.121723910
[33,] -0.280358301  0.312570223  0.2574773919 -0.12147582  0.025903145
[34,]  0.217522209  0.162182694  0.0288784029 -0.13239506 -0.154629119
[35,]  0.085453901 -0.123426653 -0.0364842031  0.11675320 -0.146860929
[36,]  0.033599765 -0.033875153  0.0099754209 -0.01037691 -0.033045458
[37,]  0.019023596 -0.038860107 -0.1789720343 -0.16766914  0.048628418
[38,]  0.110367145 -0.089575154  0.0705866644  0.10788182  0.098150664
[39,] -0.005227491 -0.184687497  0.3845820989 -0.13281485 -0.109616137
[40,] -0.183033890  0.236575232  0.0447496103  0.07202417 -0.064480317
[41,]  0.146908007  0.166834761 -0.0289345417 -0.07205531  0.439876903
[42,] -0.029867708  0.065401387 -0.1342871897  0.22571702  0.021267986
[43,] -0.053386267 -0.130519375  0.0086604449  0.24408048  0.074646340
[44,]  0.085158019 -0.177394945 -0.0393644435 -0.10746581 -0.111296707
[45,]  0.212777284  0.274544518 -0.1057275459 -0.11536478  0.035829660
[46,]  0.025280277  0.017616556 -0.0230130000  0.06754158  0.042870446
[47,]  0.042782277 -0.100193851 -0.1052509461 -0.15484204  0.078713527
[48,]  0.156185738  0.230712766  0.0088491963  0.04239920  0.032921865
[49,]  0.201622460 -0.055359892  0.0596865630  0.01179390  0.196157577
[50,]  0.017802529  0.059550133  0.0835303192 -0.02979755  0.062944469
              [,6]         [,7]         [,8]         [,9]       [,10]
 [1,]  0.000000000  0.000000000  0.540062243  0.000000000  0.00000000
 [2,]  0.066825591 -0.426811172 -0.300342946  0.031115623 -0.08596027
 [3,]  0.230062048  0.169958584 -0.027192483 -0.111304236  0.07673832
 [4,] -0.212926154  0.236624094 -0.021070235  0.005711365  0.12340282
 [5,] -0.168762666 -0.147173376  0.055671681 -0.179542226  0.09069932
 [6,] -0.317370951  0.324788594 -0.015191109 -0.023360634 -0.15108821
 [7,]  0.091466917  0.103612235 -0.035169468 -0.183861339  0.03166646
 [8,] -0.233012200 -0.161206557  0.020625064 -0.024389111 -0.38968234
 [9,] -0.467336936  0.009645931  0.171129798 -0.082596679  0.28407949
[10,]  0.108757675 -0.023983559 -0.070150055  0.003512524  0.11626733
[11,] -0.112544826 -0.165808766 -0.141305611  0.054516769  0.15962731
[12,] -0.047379465  0.125508151  0.012052936  0.230672144 -0.08088609
[13,]  0.059603217 -0.154290442  0.156409608 -0.003270297 -0.13765799
[14,] -0.052991865  0.138027292  0.024433979  0.240171751 -0.03558423
[15,] -0.137313476  0.021860797  0.155993616  0.062698798 -0.13492252
[16,]  0.059181042  0.092380826 -0.052958754 -0.099124657  0.45159574
[17,]  0.003451225  0.011409634 -0.001308639 -0.074949180 -0.03605887
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[10,] -0.0398271869 -0.3317786017  0.079999473  0.5000792960 -0.0432596328
[11,]  0.1286039858  0.3361487640  0.056586516 -0.1125650162  0.0715060491
[12,]  0.0211111636  0.2997851896  0.127503233  0.3604348973  0.0462710844
[13,] -0.1162610340  0.0525712807  0.170443963 -0.1792511137  0.0493161116
[14,] -0.0094456111 -0.0328542001  0.060044006 -0.0311143179  0.1056497929
[15,] -0.0740251780 -0.0517052237 -0.280140090  0.0609761910  0.0013469433
[16,]  0.0467971099  0.1546203686 -0.062540627  0.0192597406  0.1420136650
[17,] -0.0345312502  0.2683473911 -0.433110951 -0.0193455295  0.0962735570
[18,] -0.0211031914  0.0330222749 -0.024285838  0.1206563506  0.0236465243
[19,]  0.0149347877  0.0911611769  0.102190821  0.1458049426  0.0531377200
[20,] -0.0469688822 -0.3194644515 -0.106895230 -0.1172391625  0.0864624494
[21,]  0.0115489999  0.0348063239  0.079690698  0.0677560558 -0.0781668234
[22,] -0.0464731202 -0.1543150727 -0.088134844 -0.1191827914  0.0685667040
[23,] -0.0895180186 -0.1741747098  0.120552738 -0.1517510524 -0.0347607221
[24,] -0.1178483723  0.0239040239  0.096362171  0.0759655565 -0.0393067413
[25,] -0.0107502387  0.0398755171  0.014713927 -0.0451134428  0.1036826542
[26,]  0.0393166326 -0.0661265758  0.225905092 -0.2453017952  0.1311237522
[27,] -0.0543864143  0.0186253253  0.096002363 -0.0501326455  0.0346942245
[28,] -0.1163068784 -0.0130875880 -0.199737805 -0.0908527639  0.2006954951
[29,]  0.0710193192 -0.2228199552 -0.236124237 -0.0991740233  0.0009478463
[30,]  0.0321095229  0.0575019123  0.051325062 -0.0134053735  0.0144057651
[31,] -0.0209516952  0.2844160374  0.142928488  0.0819719454  0.0764077176
[32,]  0.0551518204 -0.0579171065 -0.230572722 -0.0521877601 -0.0782633085
[33,]  0.0307627751 -0.0560067632  0.017318537 -0.1966953146  0.0761690969
[34,] -0.2127758033 -0.1368821274  0.017690463  0.0556498532  0.1334233289
[35,]  0.0014590122 -0.0023566091 -0.067492034 -0.1552376640  0.0010798366
[36,] -0.0072580407  0.0015870577  0.006225274 -0.0521355571  0.0184640158
[37,]  0.0273462158 -0.0092251085 -0.022632633 -0.0298344966  0.1060913618
[38,] -0.0326818785 -0.1050270066  0.080543028 -0.1216364632  0.0245801847
[39,]  0.0539941362 -0.0092432674 -0.157849668 -0.0055948531 -0.1358458051
[40,] -0.0260525347  0.0542761648  0.014037211  0.0112379371  0.0122255546
[41,] -0.0809302706 -0.1110143654 -0.151479110  0.0996987707 -0.1220957992
[42,] -0.0653198823  0.0380370270  0.032542294 -0.0607965425  0.0163891042
[43,] -0.0189065016  0.0758636023 -0.001858895 -0.0144578684  0.0328494445
[44,]  0.0563229784 -0.0173927767  0.154761833  0.1811307281 -0.0398439950
[45,]  0.0002554121  0.0663085238  0.259826316 -0.1714070220 -0.3229266216
[46,] -0.0082016898 -0.0425371985  0.042808394 -0.0897821892  0.0414007859
[47,] -0.0123699916  0.0645691163 -0.030352000  0.0874466672  0.0103539264
[48,]  0.0052502741  0.1387479735 -0.269113897  0.1086692628 -0.3283622330
[49,]  0.3069925718 -0.2466812660  0.032270603 -0.1106066634 -0.0404484522
[50,]  0.0162152939 -0.0858903518 -0.030315436 -0.0373819581 -0.0721929572

Output ini menampilkan hasil Eigen Values dan Eigen Vectors dari data yang telah dianalisis sebelumnya. Dapat diketahui bahwa :

1. Interpretasi Eigenvalues

Hasil eigenvalues menunjukkan daftar nilai eigen dari matriks B, yaitu matriks hasil transformasi double-centering dari matriks jarak kuadrat. Pada MDS (khususnya Classical MDS), eigenvalues sangat penting karena menentukan jumlah dimensi yang relevan dan proporsi variasi yang dapat dijelaskan oleh masing-masing dimensi.

Pada hasil yang ditampilkan ada 2 eigenvalue besar dan positif, yaitu: λ₁ ≈ 343544.6 λ₂ ≈ 9897.626

Setelah itu, hampir semua eigenvalues lainnya mendekati 0, dengan kisaran 10⁻¹¹ sampai 10⁻¹² (nilai yang secara numerik tidak signifikan).

Makna dari pola ini :

  • Adanya dua eigenvalue positif besar menunjukkan bahwa struktur jarak antar objek dapat direpresentasikan dengan baik dalam 2 dimensi.
  • Eigenvalues kecil dan mendekati nol menunjukkan dimensi tambahan tidak memberikan informasi varians yang berarti (hanya noise numerik).
  • Tidak ada eigenvalue negatif besar, sehingga struktur data sangat cocok dengan metode Classical MDS (karena matriks B mendekati semi-definit positif).

Implikasi untuk MDS: - MDS 2 dimensi tepat untuk memvisualisasikan data. - Dua dimensi pertama mencakup hampir seluruh variabilitas (≈ 100% karena dua nilai pertama sangat dominan).

2. Interpretasi Eigenvectors

Eigenvectors menunjukkan arah atau sumbu baru dalam ruang berdimensi rendah. Dalam konteks MDS kolom eigenvectors mewakili koefisien pembentuk koordinat MDS.

Pada hasil analisis, ditampilkan eigenvectors untuk beberapa kolom:

  • Kolom [ ,1 ] mewakili dimensi pertama (Dimensi MDS 1)
  • Kolom [ ,2 ] mewakili dimensi kedua (Dimensi MDS 2)

Kolom berikutnya sebenarnya tidak penting, karena eigenvalues ke-3 dan seterusnya sangat kecil, sehingga dimensinya tidak menambah informasi. Nilai positif atau negatif tidak menjadi masalah; yang penting adalah pola relatif antar objek. Contoh:

  • Pada kolom [ ,1 ], beberapa objek memiliki nilai sangat besar atau sangat kecil → berarti objek tersebut berada jauh pada arah Dimensi 1.
  • Pada kolom [ ,2 ], objek dengan nilai tinggi memanjang ke arah sumbu Dimensi 2.

Jika misalnya objek 1 dan objek 7 memiliki nilai eigenvector yang mirip pada kolom 1 dan 2 → mereka akan berada berdekatan dalam peta MDS.

3.4 Menghitung Tingkat Kumulatif Varians

> print(cumulative_variance)
 [1] 0.9655342 0.9933516 0.9991511 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 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000
[43] 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000
[50] 1.0000000

Output ini menampilkan nilai cumulative variance yang menunjukkan proporsi total keragaman data yang dapat dijelaskan oleh sejumlah dimensi hasil dekomposisi eigen pada analisis MDS. Berdasarkan output, dua dimensi pertama mampu menjelaskan hampir seluruh variasi dalam data. Dimensi pertama menjelaskan sekitar 96,55% dari keseluruhan informasi struktur jarak antar-objek, sementara penambahan dimensi kedua meningkatkan proporsi variasi yang dijelaskan menjadi 99,34%. Ketika dimensi ketiga ditambahkan, total variasi yang terjelaskan mencapai 99,92%, dan mulai dari dimensi keempat hingga dimensi ke-50, nilai kumulatif variasi mencapai 100%, sehingga tidak ada tambahan informasi berarti pada dimensi-dimensi selanjutnya.

Hasil ini menunjukkan bahwa struktur jarak antar objek dalam dataset dapat direpresentasikan dengan sangat baik menggunakan dua dimensi utama, karena kedua dimensi tersebut sudah mencakup hampir seluruh variasi yang relevan. Dimensi ke-3 dan seterusnya hanya menambahkan kontribusi yang sangat kecil (mendekati nol), yang umumnya merupakan noise numerik. Oleh karena itu, representasi data dalam ruang dua dimensi sudah memadai dan optimal untuk tujuan visualisasi maupun interpretasi menggunakan teknik Classical MDS.

3.5 Titik Koordinat Objek

> fit
          [,1]        [,2]
1   -64.802164  11.4480074
2   -92.827450  17.9829427
3  -124.068216  -8.8304030
4   -18.340035  16.7039114
5  -107.422953 -22.5200698
6   -34.975986 -13.7195840
7    60.887282 -12.9325302
8   -66.731025  -1.3537978
9  -165.244370  -6.2746901
10  -40.535177   7.2902396
11  123.536106 -24.2912079
12   51.797002   9.4691910
13  -78.992097 -12.8970605
14   57.550961  -2.8462647
15  115.586790   3.3421305
16   55.789694  -3.1572339
17   62.383181  10.6732715
18  -78.277631   4.2949175
19   89.261044  11.4878272
20 -129.330136   5.0070315
21   21.266283 -19.4501790
22  -85.451527  -5.9045567
23   98.954816  -5.2096006
24  -86.856358  27.4284196
25   -7.986289  -5.2756414
26   62.483635   9.5105021
27   69.096544   0.2111959
28  -83.613578 -15.1021839
29  114.777355   4.7345584
30   10.815725 -23.1373389
31 -114.868163   0.3364531
32  -84.294231 -15.9239655
33 -164.325514  31.0966153
34  127.495597  16.1350394
35   50.086822 -12.2793244
36   19.693723  -3.3701310
37   11.150240  -3.8660682
38   64.689142  -8.9115466
39   -3.063973 -18.3739704
40 -107.281069  23.5361159
41   86.106720  16.5978586
42  -17.506264   6.5065756
43  -31.291122 -12.9849566
44   49.913397 -17.6484577
45  124.714469  27.3135591
46   14.817448   1.7526150
47   25.075839  -9.9679669
48   91.544647  22.9528778
49  118.176328  -5.5075792
50   10.434539   5.9244529

Negara bagian yang memiliki nilai koordinat yang berdekatan merepresentasikan tingkat kriminalitas yang mirip, sedangkan negara bagian yang posisinya jauh menunjukkan tingkat kriminalitas yang sangat berbeda. Misalnya, negara bagian seperti baris ke-9 (-165.24, -6.27) dan baris ke-33 (-164.33, 31.09) terletak jauh di sisi kiri plot, menunjukkan bahwa keduanya memiliki pola kriminalitas yang berbeda dari sebagian besar negara bagian lain. Sebaliknya, negara bagian seperti baris ke-35 (50.08, -12.28) dan baris ke-44 (49.91, -17.64) berada berdekatan, yang menunjukkan pola kriminalitas yang relatif serupa.

Secara umum, koordinat pada Dimensi 1 cenderung memisahkan negara bagian berdasarkan tingkat kriminalitas keseluruhan, terutama variabel UrbanPop, Assault, dan Rape, karena variabel-variabel inilah yang mendominasi struktur jarak dataset USArrests. Sementara itu, Dimensi 2 memberikan variasi tambahan yang lebih kecil namun tetap membantu mempertajam pemisahan kelompok.l

3.6 Menghitung Disparities

> disparities
           [,1]      [,2]      [,3]      [,4]      [,5]      [,6]       [,7]
 [1,]   0.00000  28.77711  62.63928  46.75846  54.50103  39.02574 128.032212
 [2,]  28.77711   0.00000  41.16966  74.49840  43.05256  65.96849 156.792810
 [3,]  62.63928  41.16966   0.00000 108.76787  21.55161  89.22628 185.000983
 [4,]  46.75846  74.49840 108.76787   0.00000  97.33595  34.67483  84.588926
 [5,]  54.50103  43.05256  21.55161  97.33595   0.00000  72.97953 168.583084
 [6,]  39.02574  65.96849  89.22628  34.67483  72.97953   0.00000  95.866499
 [7,] 128.03221 156.79281 185.00098  84.58893 168.58308  95.86650   0.000000
 [8,]  12.94630  32.47973  57.82260  51.65045  45.86768  34.07778 128.142497
 [9,] 101.99378  76.37174  41.25539 148.69062  60.06021 130.48095 226.229642
[10,]  24.62059  53.37430  85.07434  24.10895  73.22997  21.73286 103.418932
[11,] 191.69923 220.45474 248.08655 147.68019 230.96585 158.86423  63.670203
[12,] 116.61596 144.87483 176.81473  70.50919 162.40168  89.81799  24.175820
[13,]  28.17865  33.83772  45.25919  67.48993  30.01526  44.02380 139.879384
[14,] 123.18528 151.81410 181.71774  78.36870 166.14286  93.16365  10.623737
[15,] 180.57098 208.92786 239.96394 134.59172 224.50434 151.52641  57.069263
[16,] 121.47308 150.11317 179.94736  76.74426 164.35720  91.37818  11.024601
[17,] 127.18770 155.38266 187.46871  80.94817 173.02000 100.36841  23.653152
[18,]  15.25631  19.97647  47.63456  61.20865  39.60421  46.89941 140.227166
[19,] 154.06321 182.20430 214.29467 107.72743 199.60243 126.76850  37.435601
[20,]  64.84863  38.74042  14.80413 111.60475  35.18048  96.19455 191.061492
[21,]  91.44657 120.07755 145.72198  53.62629 128.72585  56.53346  40.153497
[22,]  26.97235  25.00034  38.72737  70.81734  27.54667  51.07695 146.507472
[23,] 164.60202 193.17953 223.05242 119.32428 207.10248 134.20089  38.843027
[24,]  27.23529  11.17457  51.95599  69.35057  54.01700  66.21730 153.157400
[25,]  59.22604  87.97150 116.13634  24.29611 100.92086  28.27974  69.297884
[26,] 127.30054 155.54201 187.45128  81.14315 172.89941 100.18989  22.499734
[27,] 134.36938 162.89633 193.37625  88.97845 177.97709 105.00075  15.496759
[28,]  32.53893  34.34416  40.93792  72.61035  24.93815  48.65724 144.517148
[29,] 179.70496 208.02710 239.23046 133.65442 223.86557 150.88611  56.712132
[30,]  83.15174 111.50240 135.64058  49.36987 118.24029  46.75013  51.100870
[31,]  51.28422  28.23458  12.98739  97.90594  24.03855  81.11925 176.255616
[32,]  33.60306  34.96419  40.40159  73.58352  24.05091  49.36748 145.212328
[33,] 101.44439  72.69073  56.69935 146.69325  78.18344 136.89336 229.476294
[34,] 192.35487 220.33080 252.79958 145.83674 238.07760 165.19175  72.674557
[35,] 117.31353 146.08317 174.18919  74.31193 157.84233  85.07500  10.820195
[36,]  85.78538 114.52933 143.86560  43.00621 128.55104  55.64071  42.288873
[37,]  77.48089 106.24848 135.30955  35.95553 120.03155  47.16694  50.556643
[38,] 131.08207 159.79609 188.75738  86.89071 172.64925  99.78103   5.533755
[39,]  68.56351  96.84682 121.38001  38.25985 104.44131  32.24965  64.182336
[40,]  44.16537  15.48370  36.46094  89.20306  46.05640  81.33887 172.077181
[41,] 150.99673 178.93953 211.70758 104.44681 197.44353 124.82055  38.833799
[42,]  47.55334  76.19047 107.65999  10.23137  94.48575  26.72618  80.767734
[43,]  41.47240  68.88926  92.87007  32.39073  76.72662   3.75738  92.178419
[44,] 118.34806 147.12086 174.20494  76.41084 157.41175  84.98025  11.944292
[45,] 190.17957 217.74193 251.39453 143.44740 237.42614 164.87802  75.456329
[46,]  80.20775 108.86160 139.28829  36.37252 124.62696  52.14187  48.353729
[47,]  92.39426 121.17111 149.14839  50.95417 133.09202  60.16890  35.933941
[48,] 156.76954 184.43907 217.94285 110.06222 204.09776 131.72828  47.197845
[49,] 183.76240 212.30732 242.26733 138.31149 226.23983 153.37232  57.768198
[50,]  75.43919 103.96367 135.30963  30.72740 121.24141  49.47731  53.861536
           [,8]      [,9]     [,10]     [,11]     [,12]      [,13]      [,14]
 [1,]  12.94630 101.99378  24.62059 191.69923 116.61596  28.178654 123.185280
 [2,]  32.47973  76.37174  53.37430 220.45474 144.87483  33.837724 151.814105
 [3,]  57.82260  41.25539  85.07434 248.08655 176.81473  45.259189 181.717735
 [4,]  51.65045 148.69062  24.10895 147.68019  70.50919  67.489926  78.368697
 [5,]  45.86768  60.06021  73.22997 230.96585 162.40168  30.015261 166.142863
 [6,]  34.07778 130.48095  21.73286 158.86423  89.81799  44.023796  93.163646
 [7,] 128.14250 226.22964 103.41893  63.67020  24.17582 139.879384  10.623737
 [8,]   0.00000  98.63617  27.58518 191.64474 119.02113  16.839857 124.290947
 [9,]  98.63617   0.00000 125.44477 289.34194 217.61164  86.506129 222.821708
[10,]  27.58518 125.44477   0.00000 167.08313  92.35789  43.433418  98.608514
[11,] 191.64474 289.34194 167.08313   0.00000  79.28596 202.848465  69.382454
[12,] 119.02113 217.61164  92.35789  79.28596   0.00000 132.687745  13.593325
[13,]  16.83986  86.50613  43.43342 202.84846 132.68775   0.000000 136.912473
[14,] 124.29095 222.82171  98.60851  69.38245  13.59333 136.912473   0.000000
[15,] 182.37828 280.99577 156.17188  28.75401  64.08337 195.255357  58.364833
[16,] 122.53399 221.05605  96.88978  70.96634  13.24267 135.133251   1.788508
[17,] 129.67316 228.25761 102.97394  70.44285  10.65443 143.326654  14.357166
[18,]  12.85426  87.60668  37.86113 203.82824 130.17751  17.206817 136.016186
[19,] 156.51975 255.12451 129.86408  49.54714  37.51839 170.011007  34.799362
[20,]  62.92145  37.64451  88.82431 254.55790 181.18209  53.427283 187.046033
[21,]  89.83877 186.97545  67.33848 102.38434  42.05300 100.472315  39.903231
[22,]  19.26568  79.79370  46.81433 209.79490 138.10688   9.519419 143.035186
[23,] 165.73070 264.20133 140.04893  31.11828  49.38954 178.112888  41.471250
[24,]  35.12044  85.32632  50.50939 216.65620 139.81162  41.085167 147.546705
[25,]  58.87550 157.26126  34.89028 132.88992  61.57477  71.413660  65.582261
[26,] 129.67059 228.27443 103.04273  69.78510  10.68671 143.239247  13.304923
[27,] 135.83659 234.43065 109.86003  59.69953  19.62103 148.667657  11.943558
[28,]  21.77243  82.10670  48.55069 207.35339 137.62187   5.120611 141.695570
[29,] 181.61046 280.23806 155.33356  30.31849  63.15807 194.569973  57.726329
[30,]  80.54825 176.86579  59.68880 112.72629  52.37032  90.389758  50.950073
[31,]  48.16680  50.80816  74.65754 239.67294 166.91520  38.238958 172.448496
[32,]  22.82008  81.52321  49.53538 207.99870 138.44001   6.105307 142.446778
[33,] 102.84801  37.38260 126.05868 293.14181 217.20195  96.006435 224.457767
[34,] 195.01241 293.59646 168.26340  40.61969  75.99152 208.518658  72.474423
[35,] 117.32765 215.41490  92.71092  74.42501  21.81565 129.080397  12.028964
[36,]  86.44827 184.96090  61.16505 105.92890  34.57555  99.144609  37.860862
[37,]  77.92177 176.41105  52.87575 114.22683  42.77836  90.593596  46.411927
[38,] 131.63730 229.94863 106.46434  60.82351  22.45125 143.736505   9.367031
[39,]  65.90281 162.63110  45.41743 126.73829  61.52210  76.125401  62.572197
[40,]  47.57955  65.17997  68.69456 235.72022 159.69881  46.126373 166.930009
[41,] 153.88840 252.38963 126.98347  55.43352  35.04247 167.712760  34.547146
[42,]  49.84840 148.28995  23.04224 144.36569  69.36656  64.474869  75.637707
[43,]  37.29974 134.12122  22.28309 155.23950  86.06872  47.701057  89.418727
[44,] 117.77707 215.45818  93.82368  73.92178  27.18299 128.993032  16.656449
[45,] 193.57994 291.89775 166.45834  51.61822  75.06916 207.637326  73.624397
[46,]  81.60762 180.24066  55.62893 111.79458  37.77609  94.946531  42.980260
[47,]  92.21011 190.35604  67.84284  99.49663  33.04276 104.109149  33.246837
[48,] 160.13121 258.44700 133.00526  57.05661  41.97243 174.264165  42.675127
[49,] 184.95400 283.42174 159.22665  19.53335  68.04791 197.306848  60.683752
[50,]  77.50805 176.10195  50.98801 117.06814  41.51408  91.385845  47.925804
           [,15]      [,16]      [,17]     [,18]     [,19]     [,20]     [,21]
 [1,] 180.570983 121.473080 127.187704  15.25631 154.06321  64.84863  91.44657
 [2,] 208.927855 150.113166 155.382661  19.97647 182.20430  38.74042 120.07755
 [3,] 239.963940 179.947361 187.468709  47.63456 214.29467  14.80413 145.72198
 [4,] 134.591722  76.744263  80.948170  61.20865 107.72743 111.60475  53.62629
 [5,] 224.504340 164.357195 173.020002  39.60421 199.60243  35.18048 128.72585
 [6,] 151.526405  91.378181 100.368415  46.89941 126.76850  96.19455  56.53346
 [7,]  57.069263  11.024601  23.653152 140.22717  37.43560 191.06149  40.15350
 [8,] 182.378281 122.533992 129.673161  12.85426 156.51975  62.92145  89.83877
 [9,] 280.995772 221.056048 228.257607  87.60668 255.12451  37.64451 186.97545
[10,] 156.171879  96.889785 102.973944  37.86113 129.86408  88.82431  67.33848
[11,]  28.754009  70.966338  70.442850 203.82824  49.54714 254.55790 102.38434
[12,]  64.083366  13.242666  10.654435 130.17751  37.51839 181.18209  42.05300
[13,] 195.255357 135.133251 143.326654  17.20682 170.01101  53.42728 100.47232
[14,]  58.364833   1.788508  14.357166 136.01619  34.79936 187.04603  39.90323
[15,]   0.000000  60.149267  53.706328 193.86676  27.55716 244.92258  97.03529
[16,]  60.149267   0.000000  15.321780 134.27428  36.53504 185.29977  38.17494
[17,]  53.706328  15.321780   0.000000 140.80535  26.89020 191.79703  50.97079
[18,] 193.866762 134.274280 140.805353   0.00000 167.69301  51.05747 102.33680
[19,]  27.557163  36.535039  26.890204 167.69301   0.00000 218.68723  74.70240
[20,] 244.922584 185.299775 191.797033  51.05747 218.68723   0.00000 152.56945
[21,]  97.035290  38.174939  50.970791 102.33680  74.70240 152.56945   0.00000
[22,] 201.250853 141.267938 148.761302  12.46973 175.57613  45.21499 107.57404
[23,]  18.701729  43.213886  39.871670 177.48712  19.30734 228.51345  78.98292
[24,] 203.870982 145.888239 150.177145  24.67293 176.83733  48.02854 117.84782
[25,] 123.873208  63.811156  72.154210  70.93989  98.68160 121.77874  32.50585
[26,]  53.460208  14.327609   1.167101 140.85786  26.85032 191.86663  50.37451
[27,]  46.595554  13.726565  12.430779 147.43074  23.10345 198.48463  51.71367
[28,] 200.052442 139.914096 148.254604  20.11765 174.90759  49.94381 104.96995
[29,]   1.610603  59.513229  52.729667 193.05549  26.39486 244.10764  96.58790
[30,] 108.065435  49.212422  61.663278  93.22100  85.74713 142.94393  11.08194
[31,] 230.474552 170.693614 177.552495  36.80403 204.43357  15.19747 137.56489
[32,] 200.807382 140.664478 149.069366  21.09509 175.70669  49.66225 105.61939
[33,] 281.284926 222.764520 227.626768  90.12530 254.34356  43.65023 192.35200
[34,]  17.477935  74.255830  65.341087 206.11358  38.51594 257.06670 112.03113
[35,]  67.337030  10.758034  26.038857 129.43005  45.82027 180.24778  29.69924
[36,]  96.127700  36.096599  44.940037  98.27074  71.13629 149.25913  16.15676
[37,] 104.685009  44.645082  53.256048  89.79948  79.60553 140.76032  18.57953
[38,]  52.351916  10.597749  19.720105 143.57545  31.93607 194.51788  44.68341
[39,] 120.621691  60.789005  71.603576  78.55554  97.03420 128.41268  24.35405
[40,] 223.780873 165.241062 170.151139  34.80550 196.91105  28.80084 135.54425
[41,]  32.323193  36.185436  24.452138 164.84410   6.00518 215.74843  74.18722
[42,] 133.130668  73.930283  79.998030  60.81160 106.88345 111.83393  46.65901
[43,] 147.782592  87.633627  96.615665  50.06322 123.01115  99.67628  52.95356
[44,]  68.946351  15.637341  30.945369 130.05557  48.96080 180.66963  28.70372
[45,]  25.650418  75.359763  64.514252 204.29305  38.82524 255.02204 113.52698
[46,] 100.781877  41.265380  48.395010  93.12979  75.07745 144.18432  22.16181
[47,]  91.484375  31.459927  42.636820 104.33298  67.67637 155.13045  10.21886
[48,]  31.025893  44.273633  31.641426 170.84416  11.69026 221.60262  82.07964
[49,]   9.220796  62.430891  58.092127 196.69837  33.54009 247.72970  97.90788
[50,] 105.183954  46.255455  52.165243  88.72714  79.02259 139.76769  27.58983
           [,22]     [,23]     [,24]     [,25]      [,26]     [,27]      [,28]
 [1,]  26.972350 164.60202  27.23529  59.22604 127.300544 134.36938  32.538930
 [2,]  25.000337 193.17953  11.17457  87.97150 155.542006 162.89633  34.344156
 [3,]  38.727371 223.05242  51.95599 116.13634 187.451279 193.37625  40.937916
 [4,]  70.817336 119.32428  69.35057  24.29611  81.143151  88.97845  72.610351
 [5,]  27.546667 207.10248  54.01700 100.92086 172.899411 177.97709  24.938151
 [6,]  51.076950 134.20089  66.21730  28.27974 100.189893 105.00075  48.657240
 [7,] 146.507472  38.84303 153.15740  69.29788  22.499734  15.49676 144.517148
 [8,]  19.265684 165.73070  35.12044  58.87550 129.670589 135.83659  21.772430
 [9,]  79.793702 264.20133  85.32632 157.26126 228.274433 234.43065  82.106704
[10,]  46.814326 140.04893  50.50939  34.89028 103.042735 109.86003  48.550688
[11,] 209.794899  31.11828 216.65620 132.88992  69.785097  59.69953 207.353394
[12,] 138.106882  49.38954 139.81162  61.57477  10.686713  19.62103 137.621865
[13,]   9.519419 178.11289  41.08517  71.41366 143.239247 148.66766   5.120611
[14,] 143.035186  41.47125 147.54670  65.58226  13.304923  11.94356 141.695570
[15,] 201.250853  18.70173 203.87098 123.87321  53.460208  46.59555 200.052442
[16,] 141.267938  43.21389 145.88824  63.81116  14.327609  13.72656 139.914096
[17,] 148.761302  39.87167 150.17714  72.15421   1.167101  12.43078 148.254604
[18,]  12.469725 177.48712  24.67293  70.93989 140.857859 147.43074  20.117651
[19,] 175.576130  19.30734 176.83733  98.68160  26.850316  23.10345 174.907587
[20,]  45.214987 228.51345  48.02854 121.77874 191.866631 198.48463  49.943810
[21,] 107.574043  78.98292 117.84782  32.50585  50.374510  51.71367 104.969950
[22,]   0.000000 184.40765  33.36257  77.46779 148.736129 154.66903   9.379467
[23,] 184.407652   0.00000 188.65586 106.94112  39.329739  30.34636 182.836215
[24,]  33.362567 188.65586   0.00000  85.38175 150.411054 158.31009  42.654049
[25,]  77.467791 106.94112  85.38175   0.00000  72.004446  77.27787  76.263018
[26,] 148.736129  39.32974 150.41105  72.00445   0.000000  11.41086 148.155932
[27,] 154.669030  30.34636 158.31009  77.27787  11.410857   0.00000 153.475996
[28,]   9.379467 182.83622  42.65405  76.26302 148.155932 153.47600   0.000000
[29,] 200.511335  18.68794 202.90679 123.17109  52.511358  45.90422 199.380186
[30,]  97.797508  89.94389 109.98514  25.93368  61.118364  62.78382  94.770550
[31,]  30.071393 213.89489  38.96968 107.02911 177.588917 183.96475  34.859727
[32,]  10.086024 183.56201  43.42803  77.04732 148.965278 154.23707   1.067058
[33,]  87.121712 265.77184  77.55595 160.51447 227.834042 235.45651  92.998633
[34,] 214.084612  35.63944 214.64925 137.16325  65.348601  60.53113 213.407703
[35,] 135.688178  49.37673 142.58380  58.49391  25.069454  22.74605 133.730197
[36,] 105.175790  79.28243 110.91199  27.74552  44.686545  49.53246 103.971340
[37,]  96.623272  87.81485 102.88167  19.18837  53.047621  58.08957  95.427624
[38,] 150.170777  34.46507 155.84169  72.76632  18.553602  10.13162 148.431873
[39,]  83.325838 102.86464  95.49357  13.99269  71.232245  74.51543  80.616026
[40,]  36.650813 208.22958  20.79228 103.39038 170.343102 177.91322  45.310797
[41,] 173.027717  25.31084 173.30184  96.60199  24.663348  23.61925 172.655357
[42,]  69.069493 117.04893  72.43728  15.14763  80.046284  86.83132  69.549374
[43,]  54.621255 130.47782  68.70762  24.54687  96.435215 101.25128  52.365276
[44,] 135.873404  50.59433 144.00657  59.20693  29.926911  26.20993 133.551252
[45,] 212.774973  41.48874 211.57086 136.64387  64.727317  61.86996 212.602142
[46,] 100.560925  84.42493 104.86567  23.86225  48.293376  54.30098  99.863663
[47,] 110.602034  74.03206 118.01401  33.39345  42.175277  45.18227 108.810614
[48,] 179.333201  29.12105 178.45713 103.45654  32.019367  31.95468 179.244502
[49,] 203.628241  19.22382 207.66122 126.16283  57.682049  49.41184 202.017877
[50,]  96.612955  89.21775  99.63904  21.55850  52.172484  58.93956  96.369953
           [,29]     [,30]     [,31]      [,32]     [,33]     [,34]      [,35]
 [1,] 179.704963  83.15174  51.28422  33.603059 101.44439 192.35487 117.313534
 [2,] 208.027101 111.50240  28.23458  34.964185  72.69073 220.33080 146.083175
 [3,] 239.230464 135.64058  12.98739  40.401591  56.69935 252.79958 174.189185
 [4,] 133.654423  49.36987  97.90594  73.583519 146.69325 145.83674  74.311929
 [5,] 223.865566 118.24029  24.03855  24.050913  78.18344 238.07760 157.842333
 [6,] 150.886111  46.75013  81.11925  49.367485 136.89336 165.19175  85.075000
 [7,]  56.712132  51.10087 176.25562 145.212328 229.47629  72.67456  10.820195
 [8,] 181.610462  80.54825  48.16680  22.820078 102.84801 195.01241 117.327646
 [9,] 280.238059 176.86579  50.80816  81.523209  37.38260 293.59646 215.414897
[10,] 155.333557  59.68880  74.65754  49.535383 126.05868 168.26340  92.710918
[11,]  30.318490 112.72629 239.67294 207.998701 293.14181  40.61969  74.425014
[12,]  63.158068  52.37032 166.91520 138.440009 217.20195  75.99152  21.815651
[13,] 194.569973  90.38976  38.23896   6.105307  96.00644 208.51866 129.080397
[14,]  57.726329  50.95007 172.44850 142.446778 224.45777  72.47442  12.028964
[15,]   1.610603 108.06544 230.47455 200.807382 281.28493  17.47793  67.337030
[16,]  59.513229  49.21242 170.69361 140.664478 222.76452  74.25583  10.758034
[17,]  52.729667  61.66328 177.55250 149.069366 227.62677  65.34109  26.038857
[18,] 193.055486  93.22100  36.80403  21.095087  90.12530 206.11358 129.430052
[19,]  26.394862  85.74713 204.43357 175.706687 254.34356  38.51594  45.820271
[20,] 244.107642 142.94393  15.19747  49.662253  43.65023 257.06670 180.247781
[21,]  96.587898  11.08194 137.56489 105.619393 192.35200 112.03113  29.699236
[22,] 200.511335  97.79751  30.07139  10.086024  87.12171 214.08461 135.688178
[23,]  18.687938  89.94389 213.89489 183.562007 265.77184  35.63944  49.376734
[24,] 202.906790 109.98514  38.96968  43.428030  77.55595 214.64925 142.583797
[25,] 123.171085  25.93368 107.02911  77.047315 160.51447 137.16325  58.493912
[26,]  52.511358  61.11836 177.58892 148.965278 227.83404  65.34860  25.069454
[27,]  45.904218  62.78382 183.96475 154.237068 235.45651  60.53113  22.746047
[28,] 199.380186  94.77055  34.85973   1.067058  92.99863 213.40770 133.730197
[29,]   0.000000 107.63300 229.68763 200.140627 280.34509  17.07995  66.890487
[30,] 107.633002   0.00000 127.85718  95.383103 183.34605 123.11179  40.744515
[31,] 229.687629 127.85718   0.00000  34.628984  58.24274 242.87813 165.436709
[32,] 200.140627  95.38310  34.62898   0.000000  92.82210 214.20250 134.430468
[33,] 280.345090 183.34605  58.24274  92.822096   0.00000 292.20440 218.755850
[34,]  17.079949 123.11179 242.87813 214.202499 292.20440   0.00000  82.459047
[35,]  66.890487  40.74452 165.43671 134.430468 218.75585  82.45905   0.000000
[36,]  95.428418  21.66936 134.61293 104.742986 187.21922 109.55225  31.671978
[37,] 103.983411  19.27417 126.08846  96.203117 178.92493 118.05205  39.835164
[38,]  51.913825  55.72000 179.79530 149.148313 232.48304  67.61643  14.985649
[39,] 120.085730  14.67432 113.35897  81.267197 168.67906 135.04323  53.499081
[40,] 222.852960 126.98529  24.40878  45.667196  57.54329 234.89329 161.392065
[41,]  31.028103  85.13295 201.63169 173.476664 250.85159  41.39146  46.166273
[42,] 132.295487  40.99874  97.55721  70.453968 148.86424 145.32119  70.155080
[43,] 147.139325  43.31348  84.63204  53.084530 140.14755 161.43480  81.381003
[44,]  68.617289  39.48108 165.76013 134.218707 219.71435  84.61869   5.371933
[45,]  24.668959 124.57214 241.09666 213.434112 289.06474  11.51929  84.480070
[46,] 100.004374  25.20959 129.69334 100.675649 181.53036 113.59234  37.958188
[47,]  90.898439  19.41090 140.32286 109.532123 193.80189 105.69377  25.117556
[48,]  29.523988  92.95949 207.64814 180.085313 255.99973  36.59172  54.406427
[49,]  10.791404 108.79847 233.11775 202.738324 284.86340  23.56378  68.425415
[50,] 104.349599  29.06429 125.42724  97.215705 176.56363 117.50552  43.631193
           [,36]      [,37]      [,38]     [,39]     [,40]      [,41]     [,42]
 [1,]  85.785384  77.480891 131.082072  68.56351  44.16537 150.996729  47.55334
 [2,] 114.529333 106.248479 159.796090  96.84682  15.48370 178.939531  76.19047
 [3,] 143.865596 135.309554 188.757376 121.38001  36.46094 211.707582 107.65999
 [4,]  43.006208  35.955533  86.890713  38.25985  89.20306 104.446809  10.23137
 [5,] 128.551038 120.031553 172.649255 104.44131  46.05640 197.443528  94.48575
 [6,]  55.640707  47.166942  99.781035  32.24965  81.33887 124.820547  26.72618
 [7,]  42.288873  50.556643   5.533755  64.18234 172.07718  38.833799  80.76773
 [8,]  86.448266  77.921775 131.637305  65.90281  47.57955 153.888396  49.84840
 [9,] 184.960901 176.411054 229.948631 162.63110  65.17997 252.389628 148.28995
[10,]  61.165054  52.875755 106.464337  45.41743  68.69456 126.983470  23.04224
[11,] 105.928900 114.226832  60.823508 126.73829 235.72022  55.433516 144.36569
[12,]  34.575551  42.778364  22.451253  61.52210 159.69881  35.042469  69.36656
[13,]  99.144609  90.593596 143.736505  76.12540  46.12637 167.712760  64.47487
[14,]  37.860862  46.411927   9.367031  62.57220 166.93001  34.547146  75.63771
[15,]  96.127700 104.685009  52.351916 120.62169 223.78087  32.323193 133.13067
[16,]  36.096599  44.645082  10.597749  60.78901 165.24106  36.185436  73.93028
[17,]  44.940037  53.256048  19.720105  71.60358 170.15114  24.452138  79.99803
[18,]  98.270744  89.799475 143.575447  78.55554  34.80550 164.844100  60.81160
[19,]  71.136286  79.605527  31.936074  97.03420 196.91105   6.005180 106.88345
[20,] 149.259128 140.760320 194.517883 128.41268  28.80084 215.748432 111.83393
[21,]  16.156760  18.579527  44.683414  24.35405 135.54425  74.187218  46.65901
[22,] 105.175790  96.623272 150.170777  83.32584  36.65081 173.027717  69.06949
[23,]  79.282435  87.814854  34.465066 102.86464 208.22958  25.310844 117.04893
[24,] 110.911994 102.881670 155.841688  95.49357  20.79228 173.301839  72.43728
[25,]  27.745522  19.188372  72.766325  13.99269 103.39038  96.601989  15.14763
[26,]  44.686545  53.047621  18.553602  71.23224 170.34310  24.663348  80.04628
[27,]  49.532461  58.089572  10.131615  74.51543 177.91322  23.619246  86.83132
[28,] 103.971340  95.427624 148.431873  80.61603  45.31080 172.655357  69.54937
[29,]  95.428418 103.983411  51.913825 120.08573 222.85296  31.028103 132.29549
[30,]  21.669364  19.274174  55.719998  14.67432 126.98529  85.132954  40.99874
[31,] 134.612926 126.088457 179.795303 113.35897  24.40878 201.631686  97.55721
[32,] 104.742986  96.203117 149.148313  81.26720  45.66720 173.476664  70.45397
[33,] 187.219220 178.924927 232.483044 168.67906  57.54329 250.851586 148.86424
[34,] 109.552251 118.052050  67.616434 135.04323 234.89329  41.391464 145.32119
[35,]  31.671978  39.835164  14.985649  53.49908 161.39207  46.166273  70.15508
[36,]   0.000000   8.557865  45.335362  27.25854 129.79424  69.349887  38.48881
[37,]   8.557865   0.000000  53.776118  20.31066 121.56009  77.699719  30.47601
[38,]  45.335362  53.776118   0.000000  68.41069 175.00458  33.308293  83.62896
[39,]  27.258538  20.310664  68.410686   0.00000 112.32835  95.783304  28.76841
[40,] 129.794236 121.560086 175.004583 112.32835   0.00000 193.512212  91.37571
[41,]  69.349887  77.699719  33.308293  95.78330 193.51221   0.000000 104.10324
[42,]  38.488808  30.476006  83.628962  28.76841  91.37571 104.103240   0.00000
[43,]  51.883516  43.409944  96.066663  28.73697  84.31050 121.067734  23.87346
[44,]  33.423036  41.140450  17.165554  52.98234 162.50006  49.827370  71.61617
[45,] 109.411361 117.766733  70.109188 135.70070 232.02629  40.067250 143.73471
[46,]   7.072523   6.709547  50.999119  26.92257 124.02648  72.818552  32.67143
[47,]   8.514612  15.203798  39.627387  29.36852 136.53159  66.562088  45.65792
[48,]  76.520952  84.749729  41.672049 103.24098 198.82657   8.364049 110.28410
[49,]  98.505798 107.038676  53.595392 121.92110 227.32042  38.950098 136.21346
[50,]  13.119519   9.816646  56.246500  27.79610 119.02577  76.421205  27.94687
          [,43]      [,44]     [,45]      [,46]      [,47]      [,48]
 [1,]  41.47240 118.348064 190.17957  80.207750  92.394260 156.769535
 [2,]  68.88926 147.120856 217.74193 108.861599 121.171114 184.439070
 [3,]  92.87007 174.204937 251.39453 139.288291 149.148393 217.942845
 [4,]  32.39073  76.410839 143.44740  36.372517  50.954168 110.062223
 [5,]  76.72662 157.411752 237.42614 124.626959 133.092018 204.097758
 [6,]   3.75738  84.980253 164.87802  52.141874  60.168898 131.728281
 [7,]  92.17842  11.944292  75.45633  48.353729  35.933941  47.197845
 [8,]  37.29974 117.777066 193.57994  81.607618  92.210109 160.131205
 [9,] 134.12122 215.458181 291.89775 180.240662 190.356041 258.446997
[10,]  22.28309  93.823681 166.45834  55.628934  67.842841 133.005256
[11,] 155.23950  73.921778  51.61822 111.794576  99.496630  57.056613
[12,]  86.06872  27.182988  75.06916  37.776090  33.042755  41.972432
[13,]  47.70106 128.993032 207.63733  94.946531 104.109149 174.264165
[14,]  89.41873  16.656449  73.62440  42.980260  33.246837  42.675127
[15,] 147.78259  68.946351  25.65042 100.781877  91.484375  31.025893
[16,]  87.63363  15.637341  75.35976  41.265380  31.459927  44.273633
[17,]  96.61566  30.945369  64.51425  48.395010  42.636820  31.641426
[18,]  50.06322 130.055571 204.29305  93.129786 104.332975 170.844156
[19,] 123.01115  48.960805  38.82524  75.077449  67.676375  11.690262
[20,]  99.67628 180.669630 255.02204 144.184316 155.130447 221.602624
[21,]  52.95356  28.703717 113.52698  22.161813  10.218858  82.079643
[22,]  54.62126 135.873404 212.77497 100.560925 110.602034 179.333201
[23,] 130.47782  50.594326  41.48874  84.424931  74.032055  29.121054
[24,]  68.70762 144.006565 211.57086 104.865675 118.014009 178.457135
[25,]  24.54687  59.206927 136.64387  23.862246  33.393446 103.456544
[26,]  96.43522  29.926911  64.72732  48.293376  42.175277  32.019367
[27,] 101.25128  26.209929  61.86996  54.300978  45.182274  31.954677
[28,]  52.36528 133.551252 212.60214  99.863663 108.810614 179.244502
[29,] 147.13933  68.617289  24.66896 100.004374  90.898439  29.523988
[30,]  43.31348  39.481081 124.57214  25.209593  19.410904  92.959491
[31,]  84.63204 165.760126 241.09666 129.693343 140.322859 207.648141
[32,]  53.08453 134.218707 213.43411 100.675649 109.532123 180.085313
[33,] 140.14755 219.714345 289.06474 181.530359 193.801890 255.999727
[34,] 161.43480  84.618688  11.51929 113.592338 105.693773  36.591716
[35,]  81.38100   5.371933  84.48007  37.958188  25.117556  54.406427
[36,]  51.88352  33.423036 109.41136   7.072523   8.514612  76.520952
[37,]  43.40994  41.140450 117.76673   6.709547  15.203798  84.749729
[38,]  96.06666  17.165554  70.10919  50.999119  39.627387  41.672049
[39,]  28.73697  52.982338 135.70070  26.922568  29.368519 103.240977
[40,]  84.31050 162.500059 232.02629 124.026484 136.531588 198.826571
[41,] 121.06773  49.827370  40.06725  72.818552  66.562088   8.364049
[42,]  23.87346  71.616174 143.73471  32.671433  45.657924 110.284097
[43,]   0.00000  81.338319 161.12639  48.406572  56.447644 127.984976
[44,]  81.33832   0.000000  87.27418  40.101462  25.997966  58.151779
[45,] 161.12639  87.274185   0.00000 112.830479 106.385003  33.455233
[46,]  48.40657  40.101462 112.83048   0.000000  15.575835  79.602225
[47,]  56.44764  25.997966 106.38500  15.575835   0.000000  74.174689
[48,] 127.98498  58.151779  33.45523  79.602225  74.174689   0.000000
[49,] 149.65437  69.334181  33.46602 103.613553  93.207275  38.977481
[50,]  45.81044  45.981109 116.26434   6.050960  21.608717  82.878325
           [,49]      [,50]
 [1,] 183.762402  75.439188
 [2,] 212.307322 103.963675
 [3,] 242.267332 135.309634
 [4,] 138.311488  30.727396
 [5,] 226.239829 121.241410
 [6,] 153.372319  49.477308
 [7,]  57.768198  53.861536
 [8,] 184.954003  77.508047
 [9,] 283.421736 176.101955
[10,] 159.226649  50.988012
[11,]  19.533354 117.068145
[12,]  68.047913  41.514076
[13,] 197.306848  91.385845
[14,]  60.683752  47.925804
[15,]   9.220796 105.183954
[16,]  62.430891  46.255455
[17,]  58.092127  52.165243
[18,] 196.698366  88.727136
[19,]  33.540087  79.022585
[20,] 247.729704 139.767686
[21,]  97.907880  27.589828
[22,] 203.628241  96.612955
[23,]  19.223822  89.217747
[24,] 207.661220  99.639045
[25,] 126.162830  21.558502
[26,]  57.682049  52.172484
[27,]  49.411836  58.939563
[28,] 202.017877  96.369953
[29,]  10.791404 104.349599
[30,] 108.798472  29.064292
[31,] 233.117754 125.427241
[32,] 202.738324  97.215705
[33,] 284.863402 176.563626
[34,]  23.563780 117.505520
[35,]  68.425415  43.631193
[36,]  98.505798  13.119519
[37,] 107.038676   9.816646
[38,]  53.595392  56.246500
[39,] 121.921099  27.796100
[40,] 227.320421 119.025775
[41,]  38.950098  76.421205
[42,] 136.213456  27.946867
[43,] 149.654367  45.810442
[44,]  69.334181  45.981109
[45,]  33.466019 116.264338
[46,] 103.613553   6.050960
[47,]  93.207275  21.608717
[48,]  38.977481  82.878325
[49,]   0.000000 108.346594
[50,] 108.346594   0.000000

Matriks disparities di atas menunjukkan besarnya ketidaksamaan atau jarak antar 30 objek yang dianalisis dalam prosedur Multidimensional Scaling (MDS). Setiap nilai pada matriks merepresentasikan tingkat perbedaan antara pasangan objek: semakin besar angkanya, semakin jauh atau semakin tidak mirip kedua objek tersebut; sebaliknya, nilai yang kecil menunjukkan objek yang relatif mirip atau dekat satu sama lain. Nilai diagonal bernilai nol karena setiap objek dibandingkan dengan dirinya sendiri sehingga tidak ada perbedaan. Pola pada matriks memperlihatkan bahwa beberapa objek memiliki tingkat kemiripan yang tinggi misalnya objek 1 dengan objek 8 atau objek 3 dengan objek 5 yang memiliki nilai ketidaksamaan rendah sedangkan pasangan lain memiliki jarak yang sangat besar seperti objek 7 dengan sebagian besar objek lainnya, mengindikasikan bahwa objek tersebut memiliki karakteristik yang paling berbeda. Secara keseluruhan, matriks ini menjadi dasar untuk memetakan posisi objek dalam ruang berdimensi rendah sehingga hubungan kemiripan dapat divisualisasikan dalam bentuk konfigurasi MDS.

3.7 Menghitung Stress

> stress <- sqrt(sum((dist_matrix - disparities)^2) / sum(dist_matrix^2))
> cat("Nilai Stress:", stress, "\n")
Nilai Stress: 0.01590128

Nilai stress yang diperoleh dari analisis MDS adalah 0.0159, yang menunjukkan tingkat ketepatan pemetaan yang sangat baik. Stress mengukur seberapa jauh konfigurasi titik hasil MDS mampu merepresentasikan struktur jarak asli dalam data; semakin kecil nilainya, semakin akurat pemetaan yang dihasilkan. Berdasarkan standar umum seperti Kruskal’s Stress Index, nilai stress di bawah 0.05 dikategorikan sebagai “excellent fit”, artinya hampir seluruh informasi dissimilarity dalam data dapat direpresentasikan dengan sangat baik dalam ruang dua dimensi. Dengan demikian, nilai stress sebesar 0.0159 mengindikasikan bahwa konfigurasi MDS yang digunakan sudah sangat optimal, distorsi jarak sangat minimal, dan visualisasi dua dimensi yang diperoleh dapat diandalkan untuk interpretasi kemiripan dan pengelompokan antar objek.

3.8 Visualisasi

> Data <- read_excel("D:/MATERI KULIAH/SMT 5/data_anmul.xlsx")
> plot(fit, type="n",
+      xlab="Dimensi 1", ylab="Dimensi 2",
+      main="Pemetaan Amerika Serikat dengan Indikator Kriminalitas")
> 
> points(fit, pch=19, cex=0.8)
> 
> text(fit,
+      labels = Data[[1]],  
+      cex = 0.55,
+      pos = 3,
+      offset = 0.6)

Plot ini menampilkan pemetaan 50 negara bagian Amerika Serikat berdasarkan kemiripan pola kriminalitas mereka. Setiap titik mewakili satu negara bagian. Negara bagian yang berdekatan memiliki karakteristik tingkat kriminalitas yang mirip, sedangkan negara bagian yang berjauhan memiliki tingkat kriminalitas yang berbeda secara signifikan. Interpretasi dilakukan berdasarkan kelompok (cluster) alami, arah dimensi, serta posisi ekstrem dari negara bagian.

1. Interpretasi Dimensi

  1. Dimensi 1 (Sumbu X) : Dimensi 1 memisahkan negara bagian berdasarkan tingkat kriminalitas keseluruhan.

  • Nilai Dimensi 1 negatif besar (kiri) → negara bagian dengan pola kriminalitas lebih rendah atau berbeda dari sebagian besar negara bagian lain. Contoh: California, Florida, Arizona, New York, Illinois berada di area kiri; ini menunjukkan karakteristik kriminalitas yang berbeda dari negara bagian di bagian tengah/kanan.

  • Nilai Dimensi 1 positif besar (kanan) → negara bagian dengan pola kriminalitas yang lebih tinggi pada indikator tertentu, atau karakteristik kriminalitas yang kontras dengan bagian kiri. Contoh: Vermont, West Virginia, South Dakota, North Dakota berada di kanan.

Kesimpulan Dimensi 1: Sumbu X berfungsi sebagai pemisah utama tipe kriminalitas: negara bagian kiri dan kanan memiliki pola kriminalitas yang sangat berbeda.

  1. Dimensi 2 (Sumbu Y) : Dimensi 2 menunjukkan jenis kriminalitas yang berbeda, bukan besarannya saja.

  • Nilai Dimensi 2 tinggi (atas) → negara bagian dengan karakteristik kriminalitas tertentu yang dominan (misalnya kriminalitas kekerasan lebih tinggi). Contoh: Mississippi, South Carolina, Alaska berada di bagian atas.

  • Nilai Dimensi 2 rendah (bawah) → negara bagian dengan pola kriminalitas yang berbeda (misalnya kriminalitas properti lebih menonjol atau tingkat kekerasan lebih rendah). Contoh: Massachusetts, New Jersey, Rhode Island berada di bawah.

Kesimpulan Dimensi 2: Sumbu Y menggambarkan perbedaan tipe kejahatan yang lebih spesifik (misal kekerasan vs properti).

2. Analisis Kelompok (Clustering Alami)

  1. Kelompok 1 — Barat & Selatan dengan kriminalitas khas (kiri-atas dan kiri-tengah) : California, Arizona, Florida, New Mexico, South Carolina, Mississippi. Negara-negara ini memiliki karakteristik kriminalitas yang cenderung lebih kompleks, banyak yang mencakup populasi besar, keberagaman wilayah, urbanisasi tinggi

  2. Kelompok 2 — Negara-negara Timur Laut & Mid-Atlantic (bawah-tengah) : Massachusetts, New Jersey, Rhode Island, Connecticut. Negara-negara ini memiliki kriminalitas yang relatif rendah, karakteristik sosial-ekonomi relatif mirip

  3. Kelompok 3 — Midwest & Great Plains (kanan-tengah dan kanan-atas) : Iowa, North Dakota, South Dakota, Nebraska. Negara-negara ini cenderung memiliki kriminalitas rendah, homogenitas demografis, dan wilayah rural yang dominan

  4. Kelompok 4 — Selatan & Tenggara dengan pola menengah (tengah) : Georgia, Tennessee, Alabama, Kentucky. Negara-negara ini cenderung memiliki pola kriminalitas yang mirip, biasanya terkait indikator seperti kekerasan antarpribadi, dan faktor sosial ekonomi menengah

3. Negara Bagian Ekstrem (Paling Menonjol)

  1. Paling Kiri (Dimensi 1 sangat negatif) yaitu California, Florida, Arizona : pola kriminalitas yang sangat berbeda dibanding negara bagian lain.

  2. Paling Kanan yaitu Vermont, North Dakota, South Dakota : karakteristik kriminalitas yang sangat kontras dengan negara bagian di sisi kiri.

  3. Paling Atas (Dimensi 2 positif tinggi) yaitu Mississippi, South Carolina, Alaska : kemungkinan memiliki tingkat kriminalitas kekerasan yang lebih tinggi.

  4. Paling Bawah yaitu Massachusetts, New Jersey, Rhode Island : pola kriminalitas lebih rendah/lebih homogen dalam kelompoknya.

4 Kesimpulan

Berdasarkan analisis jarak dan pemetaan menggunakan Multidimensional Scaling (MDS), diperoleh gambaran yang jelas mengenai kemiripan pola kriminalitas antar negara bagian di Amerika Serikat. MDS berhasil mereduksi data berdimensi tinggi menjadi dua dimensi yang tetap mempertahankan struktur kedekatan antar wilayah. Hasil pemetaan menunjukkan adanya pengelompokan alami, di mana negara-negara bagian yang memiliki karakteristik kriminalitas serupa cenderung berdekatan dalam ruang MDS. Beberapa kelompok utama yang terlihat antara lain kelompok negara bagian Barat dan Selatan dengan karakteristik kriminalitas yang lebih kompleks, kelompok negara bagian Timur Laut dengan tingkat kriminalitas yang relatif rendah dan homogen, serta kelompok negara bagian Midwest dan Great Plains yang menunjukkan pola kriminalitas yang paling berbeda dari negara bagian di pesisir.

Nilai stress yang diperoleh sebesar 0.0159, yang termasuk kategori sangat baik, menunjukkan bahwa model MDS yang digunakan mampu merepresentasikan hubungan antar negara bagian dengan distorsi yang sangat kecil. Hal ini menegaskan bahwa konfigurasi dua dimensi sudah cukup untuk menggambarkan pola kemiripan kriminalitas secara akurat. Melalui interpretasi dimensi, diketahui bahwa Dimensi 1 memisahkan negara bagian berdasarkan tingkat kriminalitas keseluruhan, sedangkan Dimensi 2 membedakan negara bagian berdasarkan jenis kriminalitas yang lebih dominan. Secara keseluruhan, hasil analisis menunjukkan bahwa MDS merupakan metode yang efektif untuk memahami struktur hubungan antar wilayah berbasis indikator kriminalitas dan dapat menjadi dasar yang kuat untuk analisis lanjutan maupun pengambilan kebijakan terkait keamanan dan kriminalitas.

5 Daftar Pustaka

U.S. Department of Commerce. (1975). Statistical Abstract of the United States 1975. U.S. Government Printing Office.

R Core Team. (2024). R: A language and environment for statistical computing. R Foundation for Statistical Computing. https://stat.ethz.ch/R-manual/R-devel/library/datasets/html/USArrests.html

Maindonald, J., & Braun, J. (2010). Data Analysis and Graphics Using R (3rd ed.). Cambridge University Press.

Hair, J. F., Black, W. C., Babin, B. J., & Anderson, R. E. (2019). Multivariate data analysis (8th ed.). Cengage Learning.

Anisa, R., & Ispriyanti, D. (2015). Penerapan analisis Multidimensional Scaling (MDS) untuk mengelompokkan kecamatan di Kabupaten Magelang. Media Statistika, 8(2), 119–130.

Federal Bureau of Investigation. (1975). Uniform Crime Reports for the United States. U.S. Government Printing Office.

Osgood, D. W. (2000). Poisson-based regression analysis of aggregate crime rates. Journal of Quantitative Criminology, 16(1), 21–43. https://doi.org/10.1023/A:1007521427059