Kemiripan Biaya Hidup di Berbagai Negara dengan Multidimensional Scaling (MDS)
Zalfa Nida Sasikirana
2024-11-24
1 PENDAHULUAN
1.1 Latar Belakang Kasus
Biaya hidup adalah salah satu faktor penting yang memengaruhi kualitas hidup seseorang, terutama bagi mereka yang memutuskan untuk tinggal atau bekerja di luar negeri. Setiap negara, bahkan setiap kota, memiliki tingkat biaya hidup yang berbeda-beda, yang dipengaruhi oleh berbagai faktor seperti inflasi, gaji rata-rata, harga barang dan jasa, serta faktor-faktor ekonomi dan sosial lainnya.
Dalam dunia yang semakin terhubung, perbandingan biaya hidup antara negara atau kota menjadi sangat penting, terutama bagi para profesional yang merencanakan untuk pindah ke negara lain, perusahaan yang ingin mengalokasikan sumber daya di berbagai lokasi, serta individu yang mempertimbangkan untuk menetap atau berinvestasi di negara lain. Mengingat pentingnya informasi ini, banyak orang yang mencari cara untuk memahami dan membandingkan biaya hidup di berbagai belahan dunia secara lebih mudah dan akurat.
1.2 Data
Dataset yang digunakan adalah data Cost Of Living Index by Country (Kaggle,2024)
Indeks biaya hidup yang disediakan di situs web ini adalah relatif terhadap New York City (NYC), yang digunakan sebagai patokan dengan indeks dasar sebesar 100%.
- X1 : Biaya Hidup tanpa sewa
- X2 : Biaya Sewa
- X3 : Biaya Bahan Makanan
- X4 : Biaya Makan di Restoran
- X5 : Kemampuan Daya Beli Lokal
> library(readxl)
> Data_Cost_Of_Living <- read_excel("D:/College 5/Anmul/Data Cost Of Living.xlsx")
> Tabel1= kable(Data_Cost_Of_Living, caption="Tabel 1 Cost Of Living dari Berbagai Negara")
> Tabel1| Country | Cost of Living Index | Rent Index | Groceries Index | Restaurant Price Index | Local Purchasing Power Index |
|---|---|---|---|---|---|
| Switzerland | 101.1 | 46.5 | 109.1 | 97.0 | 158.7 |
| Bahamas | 85.0 | 36.7 | 81.6 | 83.3 | 54.6 |
| Iceland | 83.0 | 39.2 | 88.4 | 86.8 | 120.3 |
| Singapore | 76.7 | 67.2 | 74.6 | 50.4 | 111.1 |
| Barbados | 76.6 | 19.0 | 80.8 | 69.4 | 43.5 |
| Norway | 76.0 | 26.2 | 79.0 | 73.5 | 114.7 |
| Denmark | 72.3 | 26.4 | 64.8 | 81.3 | 127.2 |
| Hong Kong (China) | 70.8 | 59.4 | 84.6 | 46.2 | 109.3 |
| United States | 70.4 | 41.7 | 75.0 | 67.2 | 142.3 |
| Australia | 70.2 | 33.4 | 77.3 | 62.5 | 127.4 |
| Austria | 65.1 | 22.5 | 66.4 | 59.3 | 102.4 |
| Canada | 64.8 | 33.2 | 71.2 | 61.7 | 103.7 |
| New Zealand | 64.6 | 25.9 | 71.9 | 57.1 | 121.0 |
| Ireland | 64.4 | 42.3 | 59.3 | 65.3 | 101.1 |
| France | 63.7 | 21.0 | 69.5 | 56.3 | 102.4 |
| Puerto Rico | 63.3 | 19.5 | 64.1 | 53.1 | 103.5 |
| Finland | 63.2 | 19.7 | 63.0 | 63.6 | 118.0 |
| Netherlands | 63.1 | 33.5 | 59.9 | 61.2 | 124.9 |
| Israel | 62.7 | 27.1 | 59.6 | 68.0 | 99.4 |
| Luxembourg | 62.4 | 40.5 | 64.1 | 68.7 | 182.5 |
| Germany | 62.2 | 24.4 | 60.8 | 52.8 | 120.2 |
| United Kingdom | 62.0 | 31.1 | 56.9 | 62.7 | 115.2 |
| Belgium | 61.1 | 20.1 | 57.5 | 66.0 | 109.8 |
| South Korea | 60.1 | 16.0 | 81.4 | 31.4 | 109.4 |
| Sweden | 59.3 | 20.9 | 60.8 | 54.7 | 122.6 |
| Italy | 56.2 | 19.2 | 57.0 | 53.3 | 78.3 |
| United Arab Emirates | 55.8 | 41.3 | 45.3 | 49.2 | 127.9 |
| Cyprus | 54.7 | 24.8 | 49.6 | 53.2 | 71.4 |
| Uruguay | 54.1 | 13.8 | 52.1 | 50.2 | 56.3 |
| Jamaica | 53.9 | 14.1 | 62.0 | 38.8 | 32.7 |
| Malta | 52.8 | 24.2 | 50.6 | 56.2 | 68.1 |
| Trinidad And Tobago | 52.6 | 13.1 | 55.3 | 43.9 | 43.7 |
| Costa Rica | 52.3 | 16.7 | 56.1 | 43.1 | 46.2 |
| Bahrain | 52.3 | 20.5 | 46.9 | 44.7 | 105.0 |
| Greece | 52.0 | 12.9 | 48.4 | 51.3 | 53.0 |
| Estonia | 52.0 | 15.0 | 46.4 | 50.6 | 80.4 |
| Qatar | 51.3 | 36.1 | 43.9 | 47.2 | 161.1 |
| Slovenia | 49.9 | 18.7 | 49.0 | 43.4 | 80.9 |
| Latvia | 49.1 | 11.2 | 42.8 | 44.9 | 66.9 |
| Spain | 47.3 | 22.9 | 46.0 | 44.1 | 92.1 |
| Lithuania | 47.1 | 15.6 | 44.2 | 43.5 | 75.5 |
| Slovakia | 46.7 | 14.8 | 46.7 | 34.5 | 68.3 |
| Cuba | 46.7 | 11.8 | 43.0 | 26.0 | 2.3 |
| Czech Republic | 46.6 | 18.8 | 44.4 | 34.1 | 84.0 |
| Panama | 46.4 | 20.1 | 48.3 | 42.2 | 42.2 |
1.3 Metode
Metode yang digunakan dalam menganalisis data ini adalah Multidimensional Scaling yang dimana untuk mengetahui kemiripan biaya hidup di berbagai negara di dunia yang ditinjau dari biaya hidup tanpa sewa, biaya sewa, biaya bahan makanan, biaya makan di restoran dan kemampuan daya beli lokal.
1.4 Tinjauan Pustaka
1.4.1 Multidimensional Scaling
Menurut Gudono (2014), multidimensional scaling (MDS) merupakan teknik algoritma yang berguna untuk mengidentifikasi dimensi mendasari evaluasi atas objek atau untuk menentukan fitur dasar objek yang diamati. Sedangkan menurut Sarwono (2013), MDS atau disebut juga perceptual mapping adalah suatu prosedur yang memungkinkan seorang peneliti menentukan citra relatif yang dilihat pada seperangkat objek, misalnya produk, perusahaan, atau hal lain yang berhubungan dengan persepsi. Perbedaan persepsi di antara semua objek direfleksikan di dalam jarak relatif di antara objek-objek tersebut di dalam suatu ruangan multidimensi.
1.4.2 Matriks Jarak
Penggunaan matriks jarak dilakukan dengan Jarak Euclidean (Euclidean Distance) yang merupakan jarak yang paling umum dan sering digunakan.
\[ d(\mathbf{x}, \mathbf{y}) = \sqrt{\sum_{i=1}^n (x_i - y_i)^2} \]
di mana x dan y adalah dua vektor dalam ruang n-dimensi dan \(d(\mathbf{x}, \mathbf{y})\) adalah jarak Euclidean antara keduanya.
1.4.3 Transformasi Matriks Jarak
Pada tahapan ini dilakukan transformasi matriks jarak ke dalam bentuk yang dapat diolah untuk menentukan koordinat objek di ruang dimensi rendah.
Matriks Centering \(J\)
Matriks \(J\) dihitung sebagai:
\[ J = I - \frac{1}{n} 1 1^T \]
Dimana: - \(I\) adalah matriks identitas berukuran \(n \times n\). - \(1\) adalah vektor kolom dengan semua elemen bernilai 1 dan berukuran \(n \times 1\).
Matriks Centroid \(B\)
Matriks \(B\) dihitung menggunakan formula:
\[ B = -\frac{1}{2} J \cdot D^2 \cdot J \]
Dimana: - \(D^2\) adalah matriks jarak yang dikuadratkan.
1.4.4 Evaluasi Model
Evaluasi model MDS dilakukan dengan memeriksa fit atau kesesuaian model dengan data asli. Biasanya pemeriksaan kualitas hasil dengan menggunakan berbagai metrik seperti Stress.
Stress adalah salah satu metrik untuk mengevaluasi seberapa baik hasil MDS merepresentasikan jarak atau dissimilaritas dalam dimensi rendah. Stress mengukur kesalahan antara jarak yang dihasilkan dari konfigurasi dalam ruang berdimensi rendah dan jarak asli dari data (Johnson & Wichern, 2002).
Semakin kecil nilai stress, semakin baik representasi dimensi rendah terhadap data asli.
Rumus Stress dapat dituliskan sebagai berikut:
\[ \text{STRESS} = \sqrt{\frac{\sum_{i < j} (d_{ij} - \hat{d}_{ij})^2}{\sum_{i < j} d_{ij}^2}} \]
Dimana: - \(d_{ij}\) adalah jarak asli antara objek \(i\) dan \(j\). - \(\hat{d}_{ij}\) adalah jarak yang diproyeksikan antara objek \(i\) dan \(j\).
Rumus ini mengukur seberapa baik jarak yang diproyeksikan (dalam ruang dimensi rendah) mencerminkan jarak asli (dalam ruang dimensi tinggi).
1.5 Tujuan
Mengetahui kemiripan dari biaya hidup yang ada di berbagai negara di dunia dengan variabel biaya hidup tanpa sewa, biaya sewa, biaya bahan makanan, biaya makan di restoran dan kemampuan daya beli lokal untuk penilaiannya.
2 SOURCE CODE
2.2 Input Data
Memasukkan data yang akan digunakan yaitu data Cost of Living Index by Country.
2.3 Statistika Deskriptif
Sebelum analisis lebih lanjut, melakukan analisis statistik deskriptif pada data untuk memahami distribusi, rata-rata, standar deviasi, dan pola data. Ini membantu untuk melihat gambaran umum dan mendeteksi potensi masalah atau outlier.
> str(Data_Cost_Of_Living)
tibble [45 × 6] (S3: tbl_df/tbl/data.frame)
$ Country : chr [1:45] "Switzerland" "Bahamas" "Iceland" "Singapore" ...
$ Cost of Living Index : num [1:45] 101.1 85 83 76.7 76.6 ...
$ Rent Index : num [1:45] 46.5 36.7 39.2 67.2 19 26.2 26.4 59.4 41.7 33.4 ...
$ Groceries Index : num [1:45] 109.1 81.6 88.4 74.6 80.8 ...
$ Restaurant Price Index : num [1:45] 97 83.3 86.8 50.4 69.4 73.5 81.3 46.2 67.2 62.5 ...
$ Local Purchasing Power Index: num [1:45] 158.7 54.6 120.3 111.1 43.5 ...
> summary(Data_Cost_Of_Living)
Country Cost of Living Index Rent Index Groceries Index
Length:45 Min. : 46.40 Min. :11.2 Min. : 42.80
Class :character 1st Qu.: 52.30 1st Qu.:18.7 1st Qu.: 48.40
Mode :character Median : 61.10 Median :22.5 Median : 59.60
Mean : 60.98 Mean :26.2 Mean : 61.32
3rd Qu.: 64.80 3rd Qu.:33.4 3rd Qu.: 71.20
Max. :101.10 Max. :67.2 Max. :109.10
Restaurant Price Index Local Purchasing Power Index
Min. :26.00 Min. : 2.30
1st Qu.:44.70 1st Qu.: 68.30
Median :53.20 Median :102.40
Mean :55.42 Mean : 95.14
3rd Qu.:63.60 3rd Qu.:120.20
Max. :97.00 Max. :182.50 2.4 Menghapus Kolom ‘Country’
Menghapus kolom kategory agar semua variabel bertipe numerik.
> mds_data <- Data_Cost_Of_Living[, c("Cost of Living Index", "Rent Index", "Groceries Index", "Restaurant Price Index", "Local Purchasing Power Index")]
> mds_data
# A tibble: 45 × 5
`Cost of Living Index` `Rent Index` `Groceries Index` Restaurant Price Inde…¹
<dbl> <dbl> <dbl> <dbl>
1 101. 46.5 109. 97
2 85 36.7 81.6 83.3
3 83 39.2 88.4 86.8
4 76.7 67.2 74.6 50.4
5 76.6 19 80.8 69.4
6 76 26.2 79 73.5
7 72.3 26.4 64.8 81.3
8 70.8 59.4 84.6 46.2
9 70.4 41.7 75 67.2
10 70.2 33.4 77.3 62.5
# ℹ 35 more rows
# ℹ abbreviated name: ¹​`Restaurant Price Index`
# ℹ 1 more variable: `Local Purchasing Power Index` <dbl>2.5 Matriks Jarak
Matriks jarak digunakan untuk mengukut seberapa dekat atau mirip objek satu dengan yang lainnya. Digunakan metode jarak euclidean.
> dist_matrix <- round(dist(mds_data, method = "euclidean"),2)
> as.matrix(dist_matrix)
1 2 3 4 5 6 7 8 9 10 11
1 0.00 110.16 48.87 81.56 127.24 66.61 66.59 81.89 57.32 65.63 91.03
2 110.16 0.00 66.22 72.96 26.47 62.50 76.32 71.38 90.73 77.34 60.74
3 48.87 66.22 0.00 49.23 81.90 22.69 30.22 48.38 34.82 31.01 46.48
4 81.56 72.96 49.23 0.00 85.40 47.41 54.72 14.71 44.11 40.05 48.53
5 127.24 26.47 81.90 85.40 0.00 71.71 86.47 80.92 101.75 85.72 62.63
6 66.61 62.50 22.69 47.41 71.71 0.00 20.80 43.99 33.00 19.25 25.38
7 66.59 76.32 30.22 54.72 86.47 20.80 0.00 55.10 27.72 23.73 34.19
8 81.89 71.38 48.38 14.71 80.92 43.99 55.10 0.00 44.00 36.37 44.10
9 57.32 90.73 34.82 44.11 101.75 33.00 27.72 44.00 0.00 17.84 46.10
10 65.63 77.34 31.01 40.05 85.72 19.25 23.73 36.37 17.84 0.00 29.98
11 91.03 60.74 46.48 48.53 62.63 25.38 34.19 44.10 46.10 29.98 0.00
12 84.87 58.36 39.61 38.62 64.16 22.26 32.86 34.26 40.48 25.07 12.04
13 78.45 75.65 40.87 44.75 80.18 22.12 27.09 39.72 29.13 13.67 19.82
14 90.37 58.65 45.09 36.43 67.00 32.16 35.81 37.49 44.54 33.71 21.92
15 91.89 62.13 48.08 49.40 62.76 26.75 36.95 43.62 47.06 30.34 4.78
16 95.69 66.15 52.77 51.28 65.74 31.14 38.55 46.37 48.64 32.78 7.56
17 83.98 74.28 44.28 52.87 77.98 23.90 23.00 49.79 35.82 23.04 16.88
18 80.36 79.95 43.79 42.94 86.73 28.98 23.84 42.53 26.22 19.00 26.02
19 93.05 57.57 46.67 49.76 61.93 28.61 32.70 48.03 48.48 34.98 12.56
20 70.19 131.92 72.20 80.38 142.35 72.33 59.28 81.93 42.46 57.96 82.71
21 88.15 79.61 50.69 48.18 82.43 31.36 31.36 45.03 35.55 23.73 20.06
22 86.95 72.58 45.89 44.73 78.21 28.73 26.09 44.19 35.57 25.25 18.69
23 91.17 69.09 48.40 54.77 72.05 28.31 27.48 52.59 43.78 31.20 14.16
24 100.56 82.13 65.57 57.51 77.90 46.51 57.79 47.19 56.32 41.39 33.46
25 88.48 82.37 51.90 52.80 84.71 32.47 30.72 49.30 36.08 25.12 22.29
26 117.68 56.59 70.64 64.17 49.53 51.56 59.56 60.02 72.95 57.49 28.20
27 96.87 93.42 63.82 47.43 98.60 50.36 43.65 49.52 40.33 38.35 40.46
28 125.31 57.20 77.68 67.01 50.27 60.06 66.68 64.55 80.00 65.60 37.33
29 138.58 58.72 90.74 82.90 43.47 72.87 81.52 78.99 96.20 80.53 51.08
30 157.35 65.74 110.23 98.89 44.11 94.08 105.99 93.65 118.41 101.99 74.12
31 126.99 55.41 78.91 69.85 47.80 61.80 68.63 67.20 82.69 68.24 39.87
32 149.69 63.00 102.02 92.01 43.72 84.84 95.01 87.42 108.57 92.51 63.64
33 147.11 61.69 99.46 88.05 43.64 82.42 92.68 83.48 105.48 89.55 61.03
34 112.01 81.03 70.86 60.11 78.32 50.48 50.87 57.42 58.86 47.22 27.71
35 143.13 61.52 95.09 87.30 45.93 77.39 85.26 83.80 100.51 85.13 55.61
36 124.97 67.33 78.82 71.22 59.33 58.87 63.35 68.11 77.31 63.05 34.46
37 96.57 123.30 78.91 71.17 128.91 69.17 57.31 72.55 46.01 53.38 66.89
38 126.22 70.02 80.65 68.46 61.64 60.71 66.26 64.61 77.21 62.90 35.54
39 139.60 71.21 92.84 83.02 58.39 73.22 78.83 79.63 92.75 78.09 49.07
40 121.11 76.30 77.18 63.60 71.32 57.43 60.05 60.62 69.15 56.56 32.72
41 133.67 72.81 87.64 76.01 62.58 67.79 72.65 72.54 84.83 70.72 42.87
42 141.20 75.72 95.35 80.68 62.51 75.54 82.32 76.32 92.91 78.18 50.63
43 195.02 97.96 147.48 131.91 77.17 131.35 141.43 128.20 154.20 138.48 110.13
44 132.14 80.40 88.16 71.84 71.49 67.91 72.31 68.09 80.76 67.40 42.59
45 154.79 68.68 106.78 92.97 52.07 89.95 98.67 89.31 111.36 96.21 67.82
12 13 14 15 16 17 18 19 20 21 22
1 84.87 78.45 90.37 91.89 95.69 83.98 80.36 93.05 70.19 88.15 86.95
2 58.36 75.65 58.65 62.13 66.15 74.28 79.95 57.57 131.92 79.61 72.58
3 39.61 40.87 45.09 48.08 52.77 44.28 43.79 46.67 72.20 50.69 45.89
4 38.62 44.75 36.43 49.40 51.28 52.87 42.94 49.76 80.38 48.18 44.73
5 64.16 80.18 67.00 62.76 65.74 77.98 86.73 61.93 142.35 82.43 78.21
6 22.26 22.12 32.16 26.75 31.14 23.90 28.98 28.61 72.33 31.36 28.73
7 32.86 27.09 35.81 36.95 38.55 23.00 23.84 32.70 59.28 31.36 26.09
8 34.26 39.72 37.49 43.62 46.37 49.79 42.53 48.03 81.93 45.03 44.19
9 40.48 29.13 44.54 47.06 48.64 35.82 26.22 48.48 42.46 35.55 35.57
10 25.07 13.67 33.71 30.34 32.78 23.04 19.00 34.98 57.96 23.73 25.25
11 12.04 19.82 21.92 4.78 7.56 16.88 26.02 12.56 82.71 20.06 18.69
12 0.00 19.35 15.63 13.56 17.73 21.45 24.09 15.31 79.80 23.32 18.71
13 19.35 0.00 29.85 19.42 20.63 13.07 15.36 27.23 64.77 12.26 17.99
14 15.63 29.85 0.00 25.32 26.43 28.54 25.74 15.63 81.66 29.13 18.51
15 13.56 19.42 25.32 0.00 6.56 18.46 27.91 16.80 83.55 20.46 21.64
16 17.73 20.63 26.43 6.56 0.00 17.94 27.15 17.81 83.22 17.75 20.42
17 21.45 13.07 28.54 18.46 17.94 0.00 15.92 20.78 67.98 12.22 13.31
18 24.09 15.36 25.74 27.91 27.15 15.92 0.00 27.16 58.66 13.31 10.60
19 15.31 27.23 15.63 16.80 17.81 20.78 27.16 0.00 84.30 25.94 17.36
20 79.80 64.77 81.66 83.55 83.22 67.98 58.66 84.30 0.00 66.36 68.60
21 23.32 12.26 29.13 20.46 17.75 12.22 13.31 25.94 66.36 0.00 13.53
22 18.71 17.99 18.51 21.64 20.42 13.31 10.60 17.36 68.60 13.53 0.00
23 20.70 21.40 24.15 17.33 15.96 10.38 20.98 12.97 75.86 17.69 12.74
24 37.05 31.68 49.16 29.13 28.77 38.38 43.66 45.22 87.41 32.77 42.96
25 26.38 13.59 32.57 22.49 19.91 11.04 14.88 27.69 64.72 5.48 15.66
26 34.46 46.64 35.64 28.38 27.13 42.04 49.94 27.80 107.88 42.82 40.31
27 39.49 33.65 35.33 41.96 38.67 33.76 21.92 40.24 61.31 25.28 24.91
28 41.89 55.42 39.02 38.24 36.64 50.57 56.38 34.24 114.45 50.63 46.42
29 56.87 69.87 56.61 51.08 49.97 64.98 73.19 49.82 131.13 65.91 63.33
30 78.32 92.11 79.33 73.24 73.07 89.50 97.30 74.52 155.27 89.60 88.08
31 44.12 58.27 41.35 40.78 39.72 53.17 59.42 36.14 117.40 54.02 49.32
32 68.72 82.05 69.00 63.14 62.40 78.25 86.26 63.24 144.24 78.62 76.69
33 65.64 79.23 65.72 60.50 59.80 75.83 83.27 60.65 141.30 75.83 73.81
34 34.62 34.86 34.86 27.97 22.15 30.08 33.51 29.79 85.91 24.56 27.10
35 61.44 74.42 60.39 55.79 54.51 69.00 77.12 53.60 134.87 70.06 67.10
36 42.15 51.17 41.35 34.97 31.64 44.79 52.35 33.15 108.74 44.59 42.74
37 66.64 52.64 66.14 67.57 64.65 53.87 43.69 68.53 38.35 47.39 51.53
38 42.19 50.87 41.96 35.28 31.83 46.46 52.58 35.97 108.73 44.23 43.72
39 56.33 66.17 55.31 49.12 46.16 60.33 67.94 48.06 124.17 59.78 58.40
40 38.62 44.53 37.10 32.86 28.39 40.04 43.86 32.62 98.20 36.17 35.80
41 49.78 58.61 48.69 42.80 39.37 53.29 59.80 42.24 115.47 51.63 50.68
42 57.09 66.08 57.03 49.85 46.75 62.28 68.62 51.61 124.18 59.66 59.99
43 114.62 128.11 113.20 109.61 108.42 124.64 131.51 109.35 189.24 123.83 122.04
44 49.06 55.03 48.55 41.94 37.53 51.46 56.00 44.11 109.58 46.94 48.33
45 72.09 85.75 70.29 67.62 66.42 81.87 88.28 66.18 145.97 81.37 78.66
23 24 25 26 27 28 29 30 31 32 33
1 91.17 100.56 88.48 117.68 96.87 125.31 138.58 157.35 126.99 149.69 147.11
2 69.09 82.13 82.37 56.59 93.42 57.20 58.72 65.74 55.41 63.00 61.69
3 48.40 65.57 51.90 70.64 63.82 77.68 90.74 110.23 78.91 102.02 99.46
4 54.77 57.51 52.80 64.17 47.43 67.01 82.90 98.89 69.85 92.01 88.05
5 72.05 77.90 84.71 49.53 98.60 50.27 43.47 44.11 47.80 43.72 43.64
6 28.31 46.51 32.47 51.56 50.36 60.06 72.87 94.08 61.80 84.84 82.42
7 27.48 57.79 30.72 59.56 43.65 66.68 81.52 105.99 68.63 95.01 92.68
8 52.59 47.19 49.30 60.02 49.52 64.55 78.99 93.65 67.20 87.42 83.48
9 43.78 56.32 36.08 72.95 40.33 80.00 96.20 118.41 82.69 108.57 105.48
10 31.20 41.39 25.12 57.49 38.35 65.60 80.53 101.99 68.24 92.51 89.55
11 14.16 33.46 22.29 28.20 40.46 37.33 51.08 74.12 39.87 63.64 61.03
12 20.70 37.05 26.38 34.46 39.49 41.89 56.87 78.32 44.12 68.72 65.64
13 21.40 31.68 13.59 46.64 33.65 55.42 69.87 92.11 58.27 82.05 79.23
14 24.15 49.16 32.57 35.64 35.33 39.02 56.61 79.33 41.35 69.00 65.72
15 17.33 29.13 22.49 28.38 41.96 38.24 51.08 73.24 40.78 63.14 60.50
16 15.96 28.77 19.91 27.13 38.67 36.64 49.97 73.07 39.72 62.40 59.80
17 10.38 38.38 11.04 42.04 33.76 50.57 64.98 89.50 53.17 78.25 75.83
18 20.98 43.66 14.88 49.94 21.92 56.38 73.19 97.30 59.42 86.26 83.27
19 12.97 45.22 27.69 27.80 40.24 34.24 49.82 74.52 36.14 63.24 60.65
20 75.86 87.41 64.72 107.88 61.31 114.45 131.13 155.27 117.40 144.24 141.30
21 17.69 32.77 5.48 42.82 25.28 50.63 65.91 89.60 54.02 78.62 75.83
22 12.74 42.96 15.66 40.31 24.91 46.42 63.33 88.08 49.32 76.69 73.81
23 0.00 42.27 17.50 34.33 35.16 42.00 56.83 82.42 44.36 70.60 68.27
24 42.27 0.00 34.15 45.47 51.19 55.11 63.82 79.72 58.22 72.24 69.52
25 17.50 34.15 0.00 44.62 26.96 52.78 67.59 91.72 55.94 80.48 77.85
26 34.33 45.47 44.62 0.00 55.70 11.66 23.48 48.43 13.78 36.59 34.01
27 35.16 51.19 26.96 55.70 0.00 59.16 77.03 100.96 62.89 89.57 86.29
28 42.00 55.11 52.78 11.66 59.16 0.00 19.10 44.43 4.99 32.06 29.17
29 56.83 63.82 67.59 23.48 77.03 19.10 0.00 28.02 16.95 14.54 13.42
30 82.42 79.72 91.72 48.43 100.96 44.43 28.02 0.00 42.30 13.95 15.65
31 44.36 58.22 55.94 13.78 62.89 4.99 16.95 42.30 0.00 29.87 27.17
32 70.60 72.24 80.48 36.59 89.57 32.06 14.54 13.95 29.87 0.00 4.54
33 68.27 69.52 77.85 34.01 86.29 29.17 13.42 15.65 27.17 4.54 0.00
34 25.82 38.31 25.54 30.10 31.50 35.11 49.77 74.39 39.01 62.32 59.66
35 60.50 68.86 71.60 27.85 80.28 22.19 5.57 27.54 19.63 13.76 13.69
36 36.52 50.01 45.75 12.62 54.46 14.19 24.89 51.60 16.90 38.41 36.37
37 59.35 69.36 46.03 85.87 33.99 90.85 107.54 131.82 94.44 120.23 117.24
38 39.31 46.01 45.82 14.45 52.93 15.71 26.51 50.50 19.21 38.24 35.56
39 52.20 60.19 61.00 22.73 68.53 18.75 16.08 40.09 19.30 26.67 25.48
40 33.52 43.88 37.55 22.11 41.46 24.14 38.54 62.72 27.85 50.53 47.68
41 45.56 53.37 52.88 19.07 59.29 16.81 22.98 47.11 19.06 34.22 32.07
42 55.38 55.53 61.25 25.87 67.49 23.06 21.80 39.65 24.73 28.38 26.19
43 116.80 114.77 125.91 82.84 131.42 76.10 60.36 38.81 74.10 47.14 49.46
44 45.46 47.03 48.47 25.54 52.41 25.54 34.16 55.13 29.11 43.67 41.03
45 73.74 77.02 83.33 39.99 89.11 32.65 19.40 19.54 30.49 11.90 11.14
34 35 36 37 38 39 40 41 42 43 44
1 112.01 143.13 124.97 96.57 126.22 139.60 121.11 133.67 141.20 195.02 132.14
2 81.03 61.52 67.33 123.30 70.02 71.21 76.30 72.81 75.72 97.96 80.40
3 70.86 95.09 78.82 78.91 80.65 92.84 77.18 87.64 95.35 147.48 88.16
4 60.11 87.30 71.22 71.17 68.46 83.02 63.60 76.01 80.68 131.91 71.84
5 78.32 45.93 59.33 128.91 61.64 58.39 71.32 62.58 62.51 77.17 71.49
6 50.48 77.39 58.87 69.17 60.71 73.22 57.43 67.79 75.54 131.35 67.91
7 50.87 85.26 63.35 57.31 66.26 78.83 60.05 72.65 82.32 141.43 72.31
8 57.42 83.80 68.11 72.55 64.61 79.63 60.62 72.54 76.32 128.20 68.09
9 58.86 100.51 77.31 46.01 77.21 92.75 69.15 84.83 92.91 154.20 80.76
10 47.22 85.13 63.05 53.38 62.90 78.09 56.56 70.72 78.18 138.48 67.40
11 27.71 55.61 34.46 66.89 35.54 49.07 32.72 42.87 50.63 110.13 42.59
12 34.62 61.44 42.15 66.64 42.19 56.33 38.62 49.78 57.09 114.62 49.06
13 34.86 74.42 51.17 52.64 50.87 66.17 44.53 58.61 66.08 128.11 55.03
14 34.86 60.39 41.35 66.14 41.96 55.31 37.10 48.69 57.03 113.20 48.55
15 27.97 55.79 34.97 67.57 35.28 49.12 32.86 42.80 49.85 109.61 41.94
16 22.15 54.51 31.64 64.65 31.83 46.16 28.39 39.37 46.75 108.42 37.53
17 30.08 69.00 44.79 53.87 46.46 60.33 40.04 53.29 62.28 124.64 51.46
18 33.51 77.12 52.35 43.69 52.58 67.94 43.86 59.80 68.62 131.51 56.00
19 29.79 53.60 33.15 68.53 35.97 48.06 32.62 42.24 51.61 109.35 44.11
20 85.91 134.87 108.74 38.35 108.73 124.17 98.20 115.47 124.18 189.24 109.58
21 24.56 70.06 44.59 47.39 44.23 59.78 36.17 51.63 59.66 123.83 46.94
22 27.10 67.10 42.74 51.53 43.72 58.40 35.80 50.68 59.99 122.04 48.33
23 25.82 60.50 36.52 59.35 39.31 52.20 33.52 45.56 55.38 116.80 45.46
24 38.31 68.86 50.01 69.36 46.01 60.19 43.88 53.37 55.53 114.77 47.03
25 25.54 71.60 45.75 46.03 45.82 61.00 37.55 52.88 61.25 125.91 48.47
26 30.10 27.85 12.62 85.87 14.45 22.73 22.11 19.07 25.87 82.84 25.54
27 31.50 80.28 54.46 33.99 52.93 68.53 41.46 59.29 67.49 131.42 52.41
28 35.11 22.19 14.19 90.85 15.71 18.75 24.14 16.81 23.06 76.10 25.54
29 49.77 5.57 24.89 107.54 26.51 16.08 38.54 22.98 21.80 60.36 34.16
30 74.39 27.54 51.60 131.82 50.50 40.09 62.72 47.11 39.65 38.81 55.13
31 39.01 19.63 16.90 94.44 19.21 19.30 27.85 19.06 24.73 74.10 29.11
32 62.32 13.76 38.41 120.23 38.24 26.67 50.53 34.22 28.38 47.14 43.67
33 59.66 13.69 36.37 117.24 35.56 25.48 47.68 32.07 26.19 49.46 41.03
34 0.00 52.99 25.90 58.37 24.41 39.56 14.08 30.50 38.92 104.97 24.39
35 52.99 0.00 27.56 110.73 29.65 16.64 41.33 24.82 23.47 57.18 36.56
36 25.90 27.56 0.00 83.52 8.77 15.83 16.24 10.18 20.83 82.19 18.24
37 58.37 110.73 83.52 0.00 82.32 97.49 70.46 88.20 96.21 162.11 80.23
38 24.41 29.65 8.77 82.32 0.00 17.13 12.62 8.35 16.39 81.08 11.32
39 39.56 16.64 15.83 97.49 17.13 0.00 28.04 10.06 12.00 67.35 21.81
40 14.08 41.33 16.24 70.46 12.62 28.04 0.00 18.23 26.93 92.33 13.62
41 30.50 24.82 10.18 88.20 8.35 10.06 18.23 0.00 11.83 75.37 13.08
42 38.92 23.47 20.83 96.21 16.39 12.00 26.93 11.83 0.00 66.72 16.37
43 104.97 57.18 82.19 162.11 81.08 67.35 92.33 75.37 66.72 0.00 82.41
44 24.39 36.56 18.24 80.23 11.32 21.81 13.62 13.08 16.37 82.41 0.00
45 63.14 16.81 39.88 120.26 38.91 27.09 50.08 33.88 27.77 44.18 42.78
45
1 154.79
2 68.68
3 106.78
4 92.97
5 52.07
6 89.95
7 98.67
8 89.31
9 111.36
10 96.21
11 67.82
12 72.09
13 85.75
14 70.29
15 67.62
16 66.42
17 81.87
18 88.28
19 66.18
20 145.97
21 81.37
22 78.66
23 73.74
24 77.02
25 83.33
26 39.99
27 89.11
28 32.65
29 19.40
30 19.54
31 30.49
32 11.90
33 11.14
34 63.14
35 16.81
36 39.88
37 120.26
38 38.91
39 27.09
40 50.08
41 33.88
42 27.77
43 44.18
44 42.78
45 0.00
> summary(dist_matrix)
Min. 1st Qu. Median Mean 3rd Qu. Max.
4.54 33.23 51.98 56.67 75.62 195.02 2.6 Transformasi Matriks Jarak
Dilakukan transformasi jarak agar dapat digunakan dalam analisis Multidimensional Scaling.
Matriks \(D^2\) digunakan untuk memperbesar perbedaan jarak yang akan digunakan dalam proses MDS untuk mempertahankan struktur relatif jarak.
> #Kuadrat Matriks Jarak
> D2 <- dist_matrix^2
> head(D2)
[1] 12135.226 2388.277 6652.034 16190.018 4436.892 4434.228> #Matriks Indentitas
> I <- diag(n)
> head(I)
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14]
[1,] 1 0 0 0 0 0 0 0 0 0 0 0 0 0
[2,] 0 1 0 0 0 0 0 0 0 0 0 0 0 0
[3,] 0 0 1 0 0 0 0 0 0 0 0 0 0 0
[4,] 0 0 0 1 0 0 0 0 0 0 0 0 0 0
[5,] 0 0 0 0 1 0 0 0 0 0 0 0 0 0
[6,] 0 0 0 0 0 1 0 0 0 0 0 0 0 0
[,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25] [,26]
[1,] 0 0 0 0 0 0 0 0 0 0 0 0
[2,] 0 0 0 0 0 0 0 0 0 0 0 0
[3,] 0 0 0 0 0 0 0 0 0 0 0 0
[4,] 0 0 0 0 0 0 0 0 0 0 0 0
[5,] 0 0 0 0 0 0 0 0 0 0 0 0
[6,] 0 0 0 0 0 0 0 0 0 0 0 0
[,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34] [,35] [,36] [,37] [,38]
[1,] 0 0 0 0 0 0 0 0 0 0 0 0
[2,] 0 0 0 0 0 0 0 0 0 0 0 0
[3,] 0 0 0 0 0 0 0 0 0 0 0 0
[4,] 0 0 0 0 0 0 0 0 0 0 0 0
[5,] 0 0 0 0 0 0 0 0 0 0 0 0
[6,] 0 0 0 0 0 0 0 0 0 0 0 0
[,39] [,40] [,41] [,42] [,43] [,44] [,45]
[1,] 0 0 0 0 0 0 0
[2,] 0 0 0 0 0 0 0
[3,] 0 0 0 0 0 0 0
[4,] 0 0 0 0 0 0 0
[5,] 0 0 0 0 0 0 0
[6,] 0 0 0 0 0 0 0> #Vektor-i
> one_matrix <- matrix(rep(1,121),nrow=121,ncol=121)
> head(one_matrix)
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14]
[1,] 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[2,] 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[3,] 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[4,] 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[5,] 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[6,] 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25] [,26]
[1,] 1 1 1 1 1 1 1 1 1 1 1 1
[2,] 1 1 1 1 1 1 1 1 1 1 1 1
[3,] 1 1 1 1 1 1 1 1 1 1 1 1
[4,] 1 1 1 1 1 1 1 1 1 1 1 1
[5,] 1 1 1 1 1 1 1 1 1 1 1 1
[6,] 1 1 1 1 1 1 1 1 1 1 1 1
[,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34] [,35] [,36] [,37] [,38]
[1,] 1 1 1 1 1 1 1 1 1 1 1 1
[2,] 1 1 1 1 1 1 1 1 1 1 1 1
[3,] 1 1 1 1 1 1 1 1 1 1 1 1
[4,] 1 1 1 1 1 1 1 1 1 1 1 1
[5,] 1 1 1 1 1 1 1 1 1 1 1 1
[6,] 1 1 1 1 1 1 1 1 1 1 1 1
[,39] [,40] [,41] [,42] [,43] [,44] [,45] [,46] [,47] [,48] [,49] [,50]
[1,] 1 1 1 1 1 1 1 1 1 1 1 1
[2,] 1 1 1 1 1 1 1 1 1 1 1 1
[3,] 1 1 1 1 1 1 1 1 1 1 1 1
[4,] 1 1 1 1 1 1 1 1 1 1 1 1
[5,] 1 1 1 1 1 1 1 1 1 1 1 1
[6,] 1 1 1 1 1 1 1 1 1 1 1 1
[,51] [,52] [,53] [,54] [,55] [,56] [,57] [,58] [,59] [,60] [,61] [,62]
[1,] 1 1 1 1 1 1 1 1 1 1 1 1
[2,] 1 1 1 1 1 1 1 1 1 1 1 1
[3,] 1 1 1 1 1 1 1 1 1 1 1 1
[4,] 1 1 1 1 1 1 1 1 1 1 1 1
[5,] 1 1 1 1 1 1 1 1 1 1 1 1
[6,] 1 1 1 1 1 1 1 1 1 1 1 1
[,63] [,64] [,65] [,66] [,67] [,68] [,69] [,70] [,71] [,72] [,73] [,74]
[1,] 1 1 1 1 1 1 1 1 1 1 1 1
[2,] 1 1 1 1 1 1 1 1 1 1 1 1
[3,] 1 1 1 1 1 1 1 1 1 1 1 1
[4,] 1 1 1 1 1 1 1 1 1 1 1 1
[5,] 1 1 1 1 1 1 1 1 1 1 1 1
[6,] 1 1 1 1 1 1 1 1 1 1 1 1
[,75] [,76] [,77] [,78] [,79] [,80] [,81] [,82] [,83] [,84] [,85] [,86]
[1,] 1 1 1 1 1 1 1 1 1 1 1 1
[2,] 1 1 1 1 1 1 1 1 1 1 1 1
[3,] 1 1 1 1 1 1 1 1 1 1 1 1
[4,] 1 1 1 1 1 1 1 1 1 1 1 1
[5,] 1 1 1 1 1 1 1 1 1 1 1 1
[6,] 1 1 1 1 1 1 1 1 1 1 1 1
[,87] [,88] [,89] [,90] [,91] [,92] [,93] [,94] [,95] [,96] [,97] [,98]
[1,] 1 1 1 1 1 1 1 1 1 1 1 1
[2,] 1 1 1 1 1 1 1 1 1 1 1 1
[3,] 1 1 1 1 1 1 1 1 1 1 1 1
[4,] 1 1 1 1 1 1 1 1 1 1 1 1
[5,] 1 1 1 1 1 1 1 1 1 1 1 1
[6,] 1 1 1 1 1 1 1 1 1 1 1 1
[,99] [,100] [,101] [,102] [,103] [,104] [,105] [,106] [,107] [,108]
[1,] 1 1 1 1 1 1 1 1 1 1
[2,] 1 1 1 1 1 1 1 1 1 1
[3,] 1 1 1 1 1 1 1 1 1 1
[4,] 1 1 1 1 1 1 1 1 1 1
[5,] 1 1 1 1 1 1 1 1 1 1
[6,] 1 1 1 1 1 1 1 1 1 1
[,109] [,110] [,111] [,112] [,113] [,114] [,115] [,116] [,117] [,118]
[1,] 1 1 1 1 1 1 1 1 1 1
[2,] 1 1 1 1 1 1 1 1 1 1
[3,] 1 1 1 1 1 1 1 1 1 1
[4,] 1 1 1 1 1 1 1 1 1 1
[5,] 1 1 1 1 1 1 1 1 1 1
[6,] 1 1 1 1 1 1 1 1 1 1
[,119] [,120] [,121]
[1,] 1 1 1
[2,] 1 1 1
[3,] 1 1 1
[4,] 1 1 1
[5,] 1 1 1
[6,] 1 1 1Matriks centering \(J\) digunakan untuk mengubah matriks jarak yang dikuadratkan menjadi matriks B.
> #Matriks Centering J
> J <- I - (1/n) * D2
> head(J)
[,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] -268.67168 -118.29248 -20.29441 -52.33691 -96.03884 -18.34251 -147.67801
[2,] -53.07282 -14.57024 -52.01388 -33.14454 -135.13068 -17.81014 -249.64178
[3,] -147.82297 -86.80556 -25.94294 -44.50139 -167.15762 -48.07067 -104.66838
[4,] -359.77817 -129.43872 -21.36934 -28.49211 -85.22944 -23.42891 -200.59778
[5,] -98.59760 -113.22454 -48.00868 -54.23022 -449.30050 -59.07630 -190.87961
[6,] -98.53840 -182.93184 -34.86560 -58.43641 -150.99344 -55.35844 -57.50571
[,8] [,9] [,10] [,11] [,12] [,13] [,14]
[1,] -116.96672 -19.973342 -7.76258 -26.63432 -15.52322 -36.95574 -9.595742
[2,] -92.76560 -13.966776 -4.45568 -83.88608 -18.10070 -88.59244 -102.695120
[3,] -140.90971 -4.152642 -24.87937 -39.48050 -14.72328 -81.33889 -3.318409
[4,] -81.66188 -25.252536 -11.04098 -98.68644 -5.42882 -17.38491 -3.936802
[5,] -116.93448 -20.455902 -17.67200 -39.55547 -148.18568 -69.16720 -2.394320
[6,] -129.43872 -23.878409 -36.37804 -70.51264 -18.85682 -27.17558 -32.733876
[,15] [,16] [,17] [,18] [,19] [,20]
[1,] -42.41814 -306.2836 -89.12642 -75.32374 -163.859042 -6.3845000
[2,] -61.43681 -462.3373 -172.40192 -115.96928 -4.640056 -4.6980356
[3,] -102.57430 -443.6820 -54.05472 -107.40068 -11.481176 -4.0021422
[4,] -42.74888 -164.0117 -130.69680 -32.61458 -10.863380 -55.0456200
[5,] -79.46756 -404.2204 -121.06480 -105.37110 -8.081442 -0.6894422
[6,] -104.63788 -262.7642 -16.32022 -55.57778 -14.872376 -13.7669356
[,21] [,22] [,23] [,24] [,25] [,26]
[1,] -86.306276 -155.013120 -269.67168 -118.29248 -20.29441 -52.33691
[2,] -4.207502 -1.709176 -53.07282 -15.57024 -52.01388 -33.14454
[3,] -32.785069 -5.568642 -147.82297 -86.80556 -26.94294 -44.50139
[4,] -321.227842 -5.860836 -359.77817 -129.43872 -21.36934 -29.49211
[5,] -32.495502 -2.302942 -98.59760 -113.22454 -48.00868 -54.23022
[6,] -15.806420 -9.641976 -98.53840 -182.93184 -34.86560 -58.43641
[,27] [,28] [,29] [,30] [,31] [,32] [,33]
[1,] -96.03884 -18.34251 -147.67801 -116.96672 -19.973342 -7.76258 -26.63432
[2,] -135.13068 -17.81014 -249.64178 -92.76560 -13.966776 -4.45568 -83.88608
[3,] -167.15762 -48.07067 -104.66838 -140.90971 -4.152642 -24.87937 -39.48050
[4,] -85.22944 -23.42891 -200.59778 -81.66188 -25.252536 -11.04098 -98.68644
[5,] -450.30050 -59.07630 -190.87961 -116.93448 -20.455902 -17.67200 -39.55547
[6,] -150.99344 -56.35844 -57.50571 -129.43872 -23.878409 -36.37804 -70.51264
[,34] [,35] [,36] [,37] [,38] [,39]
[1,] -15.52322 -36.95574 -9.595742 -42.41814 -306.2836 -89.12642
[2,] -18.10070 -88.59244 -102.695120 -61.43681 -462.3373 -172.40192
[3,] -14.72328 -81.33889 -3.318409 -102.57430 -443.6820 -54.05472
[4,] -5.42882 -17.38491 -3.936802 -42.74888 -164.0117 -130.69680
[5,] -148.18568 -69.16720 -2.394320 -79.46756 -404.2204 -121.06480
[6,] -18.85682 -27.17558 -32.733876 -104.63788 -262.7642 -16.32022
[,40] [,41] [,42] [,43] [,44] [,45]
[1,] -75.32374 -163.859042 -6.3845000 -86.306276 -155.013120 -269.67168
[2,] -115.96928 -4.640056 -4.6980356 -4.207502 -1.709176 -53.07282
[3,] -107.40068 -11.481176 -4.0021422 -32.785069 -5.568642 -147.82297
[4,] -32.61458 -10.863380 -55.0456200 -321.227842 -5.860836 -359.77817
[5,] -105.37110 -8.081442 -0.6894422 -32.495502 -2.302942 -98.59760
[6,] -55.57778 -14.872376 -13.7669356 -15.806420 -9.641976 -98.53840Matriks \(B\) adalah bentuk transformasi dari matriks jarak yang dikuadratkan. Ini mempersiapkan data untuk langkah berikutnya dalam MDS, yaitu dekomposisi eigen.
> #Matriks B
> B <- round((-0.5)*J*D2*J,2)
> head(B)
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] -437987424 -37243795.10 -188066.6 -3225571.9 -19930729 -138854.1
[2,] -3363559 -74372.34 -3166213.2 -819254.0 -55519349 -127111.3
[3,] -72678889 -14717196.24 -408005.5 -1982911.0 -105089918 -2499326.6
[4,] -1047820617 -48794980.30 -219561.2 -538688.3 -13930011 -289360.1
[5,] -21566595 -32659101.95 -2489669.5 -3588448.1 -2045308160 -4638979.2
[6,] -21527771 -137736943.71 -953617.1 -4489862.9 -77456305 -3886056.1
[,7] [,8] [,9] [,10] [,11] [,12]
[1,] -72465285 -36005552 -179281.20 -10524.48 -425115.9 -84164.31
[2,] -350053424 -17961534 -61301.48 -1990.33 -13281655.7 -133434.70
[3,] -25800550 -62951382 -1611.22 -346497.89 -1384619.5 -71811.87
[4,] -181618835 -12252947 -362324.33 -30283.45 -21624939.8 -3599.97
[5,] -156480825 -35975785 -192592.09 -124176.56 -1392522.2 -73215196.16
[6,] -4278735 -48794980 -306336.45 -1083179.04 -7888301.2 -150864.79
[,13] [,14] [,15] [,16] [,17] [,18]
[1,] -1135607.6 -19880.08 -1717265 -646477714 -15929491.22 -9615639.7
[2,] -15644887.4 -24368676.26 -5217572 -2223613280 -115294559.02 -35092265.0
[3,] -12108134.2 -822.19 -24282770 -1965160146 -3553721.46 -27874273.5
[4,] -118222.4 -1372.82 -1757749 -99267547 -50231645.30 -780580.8
[5,] -7445316.2 -308.84 -11291511 -1486062660 -39924198.73 -26323709.4
[6,] -451563.4 -789177.69 -25778002 -408207471 -97804.92 -3862656.6
[,19] [,20] [,21] [,22] [,23]
[1,] -98990553.86 -5855.49 -14464707.13 -83808465.84 -441253883
[2,] -2247.77 -2333.09 -1675.93 -112.34 -3363559
[3,] -34051.92 -1442.31 -792886.13 -3885.35 -72678889
[4,] -28845.46 -3752760.26 -745799451.17 -4529.61 -1047820617
[5,] -11875.42 -7.37 -772062.18 -274.81 -21566595
[6,] -74015.65 -58707.60 -88855.25 -20168.82 -21527771
[,24] [,25] [,26] [,27] [,28] [,29]
[1,] -37243795.10 -188066.6 -3225572 -19930729 -138854.1 -72465285
[2,] -84931.47 -3166213.2 -819254 -55519349 -127111.3 -350053424
[3,] -14717196.24 -440065.8 -1982911 -105089918 -2499326.6 -25800550
[4,] -48794980.30 -219561.2 -577165 -13930011 -289360.1 -181618835
[5,] -32659101.95 -2489669.5 -3588448 -2054422703 -4638979.2 -156480825
[6,] -137736943.71 -953617.1 -4489863 -77456305 -4027720.3 -4278735
[,30] [,31] [,32] [,33] [,34] [,35]
[1,] -36005552 -179281.20 -10524.48 -425115.9 -84164.31 -1135607.6
[2,] -17961534 -61301.48 -1990.33 -13281655.7 -133434.70 -15644887.4
[3,] -62951382 -1611.22 -346497.89 -1384619.5 -71811.87 -12108134.2
[4,] -12252947 -362324.33 -30283.45 -21624939.8 -3599.97 -118222.4
[5,] -35975785 -192592.09 -124176.56 -1392522.2 -73215196.16 -7445316.2
[6,] -48794980 -306336.45 -1083179.04 -7888301.2 -150864.79 -451563.4
[,36] [,37] [,38] [,39] [,40] [,41]
[1,] -19880.08 -1717265 -646477714 -15929491.22 -9615639.7 -98990553.86
[2,] -24368676.26 -5217572 -2223613280 -115294559.02 -35092265.0 -2247.77
[3,] -822.19 -24282770 -1965160146 -3553721.46 -27874273.5 -34051.92
[4,] -1372.82 -1757749 -99267547 -50231645.30 -780580.8 -28845.46
[5,] -308.84 -11291511 -1486062660 -39924198.73 -26323709.4 -11875.42
[6,] -789177.69 -25778002 -408207471 -97804.92 -3862656.6 -74015.65
[,42] [,43] [,44] [,45]
[1,] -5855.49 -14464707.13 -83808465.84 -441253883
[2,] -2333.09 -1675.93 -112.34 -3363559
[3,] -1442.31 -792886.13 -3885.35 -72678889
[4,] -3752760.26 -745799451.17 -4529.61 -1047820617
[5,] -7.37 -772062.18 -274.81 -21566595
[6,] -58707.60 -88855.25 -20168.82 -21527771Dekomposisi eigen pada matriks B digunakan untuk mendapatkan koordinat dalam ruang berdimensi lebih rendah (digambarkan dari eigenvalue dan eigenvector).
> #Dekomposisi eigen
> eigen_result <- eigen(B)
> head(eigen_result)
$values
[1] -7.400233e+09+0.000000e+00i 1.007812e+09+3.812613e+09i
[3] 1.007812e+09-3.812613e+09i 3.549306e+09+0.000000e+00i
[5] -1.851470e+09+0.000000e+00i 1.564079e+08+9.359505e+08i
[7] 1.564079e+08-9.359505e+08i -6.544124e+08+5.461725e+08i
[9] -6.544124e+08-5.461725e+08i -6.767326e+08+3.965811e+08i
[11] -6.767326e+08-3.965811e+08i 4.388551e+08+3.165067e+08i
[13] 4.388551e+08-3.165067e+08i 2.923664e+08+1.609691e+08i
[15] 2.923664e+08-1.609691e+08i -1.557912e+08+2.412795e+08i
[17] -1.557912e+08-2.412795e+08i 1.814814e+08+0.000000e+00i
[19] -3.374183e+07+1.494517e+08i -3.374183e+07-1.494517e+08i
[21] 1.634261e+07+3.255446e+07i 1.634261e+07-3.255446e+07i
[23] 4.033200e+06+0.000000e+00i 2.376604e+06+4.074437e+05i
[25] 2.376604e+06-4.074437e+05i -7.638922e+05+1.050758e+06i
[27] -7.638922e+05-1.050758e+06i 1.039592e+06+0.000000e+00i
[29] 7.879134e+05+0.000000e+00i 4.869533e+05+5.799694e+05i
[31] 4.869533e+05-5.799694e+05i 5.539494e+05+1.310045e+05i
[33] 5.539494e+05-1.310045e+05i 4.416748e+05+0.000000e+00i
[35] -1.697232e+05+0.000000e+00i 1.401126e+05+0.000000e+00i
[37] 1.332933e+05+3.013994e+04i 1.332933e+05-3.013994e+04i
[39] 1.224348e+05+0.000000e+00i -9.448259e+03+8.368720e+04i
[41] -9.448259e+03-8.368720e+04i 7.588184e+04+0.000000e+00i
[43] 2.796401e+04+0.000000e+00i 2.466452e+04+0.000000e+00i
[45] 8.914138e+03+0.000000e+00i
$vectors
[,1] [,2] [,3]
[1,] 0.040369428+0i 0.012837615+0.0203245607i 0.012837615-0.0203245607i
[2,] 0.054234055+0i -0.024664810+0.0609574427i -0.024664810-0.0609574427i
[3,] 0.038899444+0i -0.016543040+0.0441162206i -0.016543040-0.0441162206i
[4,] 0.071226961+0i 0.052944232+0.0160729994i 0.052944232-0.0160729994i
[5,] 0.067601593+0i -0.037404977+0.0386263862i -0.037404977-0.0386263862i
[6,] 0.012559384+0i -0.006282389+0.0064370160i -0.006282389-0.0064370160i
[7,] 0.151082305+0i -0.055329062-0.0106730698i -0.055329062+0.0106730698i
[8,] 0.009497537+0i -0.004686153+0.0100944464i -0.004686153-0.0100944464i
[9,] 0.029268785+0i -0.013536805+0.0178774238i -0.013536805-0.0178774238i
[10,] 0.037204331+0i -0.012590677-0.0087947908i -0.012590677+0.0087947908i
[11,] 0.030995245+0i -0.012786021+0.0192974866i -0.012786021-0.0192974866i
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[45,] 0.256925046+0i -0.070367951-0.2422161455i -0.070367951+0.2422161455i
[,4] [,5] [,6]
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[3,] -0.052110575+0i 0.013524247+0i -0.1455107422+0.049698433i
[4,] 0.046671668+0i 0.042231660+0i 0.0778926009+0.049856375i
[5,] -0.046007236+0i -0.767717728+0i -0.0266899594-0.037486482i
[6,] -0.009661531+0i -0.011963893+0i -0.0236061918-0.012419474i
[7,] -0.154554320+0i 0.058263364+0i 0.0481068044-0.080195204i
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[10,] 0.014063829+0i 0.003711328+0i 0.0825313645-0.092237646i
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[25,] 0.005190196+0i 0.065968790+0i 0.0378898784-0.010246030i
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[28,] 0.077849007+0i 0.004601538+0i 0.0045540824+0.030131359i
[29,] 0.196820893+0i 0.061205811+0i -0.0426069398+0.067565897i
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[32,] 0.124670634+0i 0.041939682+0i -0.0377939439+0.058513505i
[33,] -0.090288844+0i 0.112443418+0i -0.1490485698-0.157316582i
[34,] 0.127291564+0i 0.074229492+0i -0.0114044448-0.056802305i
[35,] 0.034590465+0i 0.031229305+0i -0.0311412818-0.009035544i
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[27,] -0.0103479767+3.273179e-03i -0.0609971549+0i 0.0013586418+0i
[28,] -0.0470872183-1.226812e-03i -0.2016877786+0i 0.0282336287+0i
[29,] -0.0027466536-3.769364e-03i 0.0070661213+0i 0.0036099637+0i
[30,] 0.0005837159+2.368715e-04i 0.0001640579+0i -0.0006574097+0i
[31,] 0.0011766785-6.045998e-06i -0.0635755830+0i -0.0052838764+0i
[32,] 0.0058724420+8.714274e-03i 0.0308893836+0i -0.0034443496+0i
[33,] 0.0779201584-3.528623e-02i 0.2400646516+0i -0.0229874086+0i
[34,] 0.1738822319-1.803141e-01i -0.0599381847+0i 0.6604519623+0i
[35,] -0.0142462882-5.288197e-02i 0.0344611695+0i -0.0058330631+0i
[36,] 0.0264273003+2.359193e-03i 0.0140737623+0i -0.0020854611+0i
[37,] 0.5416007326+5.198158e-07i -0.3512339533+0i -0.0559996585+0i
[38,] -0.0761744280+2.918431e-02i -0.0250064429+0i 0.0649024844+0i
[39,] -0.0133406460-1.396418e-02i -0.0646715865+0i 0.0255885278+0i
[40,] 0.0046666706+1.366670e-02i 0.0312055103+0i 0.0059092643+0i
[41,] -0.0911298398+2.588670e-01i 0.1324122655+0i -0.2196949550+0i
[42,] 0.0450804264+1.716480e-02i -0.1036609277+0i 0.0008628765+0i
[43,] 0.0465419475-5.161777e-03i -0.0832097471+0i -0.0759603859+0i
[44,] -0.0817551151+3.874653e-02i -0.2357093942+0i 0.0097480236+0i
[45,] 0.0019184681+1.343301e-03i -0.0015761647+0i -0.0002294971+0i
[,44] [,45]
[1,] 8.610835e-04+0i 1.121972e-03+0i
[2,] 5.848563e-05+0i 2.144764e-03+0i
[3,] -3.141103e-02+0i 5.933809e-03+0i
[4,] 6.262198e-03+0i 6.173083e-04+0i
[5,] 7.121051e-04+0i -1.452607e-04+0i
[6,] 2.372215e-02+0i -5.293104e-03+0i
[7,] 3.670457e-03+0i -9.392512e-03+0i
[8,] -7.691131e-04+0i 3.665844e-04+0i
[9,] -4.368032e-03+0i -2.743881e-04+0i
[10,] -4.739919e-03+0i 8.722844e-04+0i
[11,] -1.406840e-02+0i 2.815575e-02+0i
[12,] 6.684883e-01+0i 9.050658e-02+0i
[13,] -6.159988e-03+0i 2.790371e-02+0i
[14,] -3.427502e-05+0i 6.076110e-03+0i
[15,] -8.805923e-03+0i 1.203092e-01+0i
[16,] 5.657674e-02+0i -1.955296e-02+0i
[17,] 2.391393e-02+0i 8.572049e-03+0i
[18,] 4.568359e-03+0i 1.215588e-03+0i
[19,] -1.973750e-01+0i 6.731720e-01+0i
[20,] 5.533682e-04+0i -1.187063e-01+0i
[21,] -9.169194e-02+0i 4.950584e-02+0i
[22,] 1.581650e-03+0i -7.374072e-02+0i
[23,] -9.273973e-04+0i 4.508692e-05+0i
[24,] -4.004347e-05+0i -2.145051e-03+0i
[25,] 3.138429e-02+0i -5.913409e-03+0i
[26,] -6.272567e-03+0i -6.409587e-04+0i
[27,] -7.085105e-04+0i 1.437159e-04+0i
[28,] -2.373502e-02+0i 5.303285e-03+0i
[29,] -3.706174e-03+0i 9.450984e-03+0i
[30,] 7.754660e-04+0i -3.698282e-04+0i
[31,] 4.389174e-03+0i 2.703981e-04+0i
[32,] 4.774721e-03+0i -8.883918e-04+0i
[33,] 1.406913e-02+0i -2.815515e-02+0i
[34,] -6.684891e-01+0i -9.050652e-02+0i
[35,] 6.141508e-03+0i -2.769126e-02+0i
[36,] 2.051539e-05+0i -6.119873e-03+0i
[37,] 8.803138e-03+0i -1.203060e-01+0i
[38,] -5.657669e-02+0i 1.955094e-02+0i
[39,] -2.379681e-02+0i -8.740079e-03+0i
[40,] -4.588500e-03+0i -1.180483e-03+0i
[41,] 1.974063e-01+0i -6.731426e-01+0i
[42,] -5.546581e-04+0i 1.186496e-01+0i
[43,] 9.169095e-02+0i -4.950757e-02+0i
[44,] -1.587674e-03+0i 7.374499e-02+0i
[45,] 6.670684e-05+0i -1.167481e-03+0iEigen value memberikan informasi tentang seberapa banyak variasi yang dijelaskan oleh masing-masing dimensi.
Eigen vector digunakan untuk menemukan koordinat objek di ruang dimensi rendah.
> #Eigenvalue dan Eigenvector
> eigenvalues <- eigen_result$values
> head(eigenvalues)
[1] -7400232981+ 0i 1007811962+3812613341i 1007811962-3812613341i
[4] 3549305504+ 0i -1851470060+ 0i 156407928+ 935950458i
> eigenvectors <- eigen_result$vectors
> head(eigenvectors)
[,1] [,2] [,3]
[1,] 0.04036943+0i 0.012837615+0.020324561i 0.012837615-0.020324561i
[2,] 0.05423405+0i -0.024664810+0.060957443i -0.024664810-0.060957443i
[3,] 0.03889944+0i -0.016543040+0.044116221i -0.016543040-0.044116221i
[4,] 0.07122696+0i 0.052944232+0.016072999i 0.052944232-0.016072999i
[5,] 0.06760159+0i -0.037404977+0.038626386i -0.037404977-0.038626386i
[6,] 0.01255938+0i -0.006282389+0.006437016i -0.006282389-0.006437016i
[,4] [,5] [,6]
[1,] 0.006700400+0i 0.01498726+0i -0.04869386+0.03455043i
[2,] -0.067709637+0i 0.06600598+0i -0.13873360+0.06453277i
[3,] -0.052110575+0i 0.01352425+0i -0.14551074+0.04969843i
[4,] 0.046671668+0i 0.04223166+0i 0.07789260+0.04985638i
[5,] -0.046007236+0i -0.76771773+0i -0.02668996-0.03748648i
[6,] -0.009661531+0i -0.01196389+0i -0.02360619-0.01241947i
[,7] [,8]
[1,] -0.04869386-0.03455043i 0.015898409-0.020684559i
[2,] -0.13873360-0.06453277i 0.104763905-0.022310491i
[3,] -0.14551074-0.04969843i 0.088692219-0.032873450i
[4,] 0.07789260-0.04985638i 0.045538503-0.029654760i
[5,] -0.02668996+0.03748648i -0.008822695-0.002183669i
[6,] -0.02360619+0.01241947i 0.024365264+0.012166983i
[,9] [,10]
[1,] 0.015898409+0.020684559i 0.012879738+1.753593e-02i
[2,] 0.104763905+0.022310491i -0.021639773+6.362655e-02i
[3,] 0.088692219+0.032873450i 0.008308768+4.856552e-02i
[4,] 0.045538503+0.029654760i 0.003521622+1.638902e-02i
[5,] -0.008822695+0.002183669i 0.052713924-4.594759e-02i
[6,] 0.024365264-0.012166983i -0.021673966+8.808376e-05i
[,11] [,12]
[1,] 0.012879738-1.753593e-02i 0.0202250941+0.01789313i
[2,] -0.021639773-6.362655e-02i -0.0265219892+0.12470740i
[3,] 0.008308768-4.856552e-02i -0.0058782423+0.03284360i
[4,] 0.003521622-1.638902e-02i -0.0009703151+0.09133854i
[5,] 0.052713924+4.594759e-02i 0.0103916748+0.02647878i
[6,] -0.021673966-8.808376e-05i 0.0157507977-0.02280028i
[,13] [,14]
[1,] 0.0202250941-0.01789313i -0.02226825-0.0093229507i
[2,] -0.0265219892-0.12470740i -0.05671897+0.0133667926i
[3,] -0.0058782423-0.03284360i -0.03197864-0.0082413607i
[4,] -0.0009703151-0.09133854i 0.02280072+0.0005885581i
[5,] 0.0103916748-0.02647878i -0.04084396+0.0048433440i
[6,] 0.0157507977+0.02280028i -0.01374735-0.0090252523i
[,15] [,16]
[1,] -0.02226825+0.0093229507i 0.03655683-3.333164e-03i
[2,] -0.05671897-0.0133667926i 0.12643616-5.473251e-02i
[3,] -0.03197864+0.0082413607i 0.03223918-3.120957e-02i
[4,] 0.02280072-0.0005885581i 0.06211027+8.398402e-03i
[5,] -0.04084396-0.0048433440i 0.06501317-7.484813e-05i
[6,] -0.01374735+0.0090252523i 0.01570677-9.009842e-03i
[,17] [,18] [,19]
[1,] 0.03655683+3.333164e-03i 0.037759507+0i 0.040159015-0.001829196i
[2,] 0.12643616+5.473251e-02i 0.064084155+0i 0.106289070-0.028376058i
[3,] 0.03223918+3.120957e-02i 0.041302924+0i 0.010072007+0.001334385i
[4,] 0.06211027-8.398402e-03i -0.008271959+0i 0.071238531-0.003143519i
[5,] 0.06501317+7.484813e-05i 0.042833529+0i 0.056886096-0.001208305i
[6,] 0.01570677+9.009842e-03i 0.009826790+0i -0.002041655-0.001947320i
[,20] [,21]
[1,] 0.040159015+0.001829196i 0.01326789-0.020803185i
[2,] 0.106289070+0.028376058i 0.02697303-0.009018432i
[3,] 0.010072007-0.001334385i -0.03387892+0.040662080i
[4,] 0.071238531+0.003143519i 0.06219272+0.013120331i
[5,] 0.056886096+0.001208305i 0.04166513+0.023707111i
[6,] -0.002041655+0.001947320i -0.01027128+0.013334092i
[,22] [,23] [,24]
[1,] 0.01326789+0.020803185i -0.284949663+0i -0.10373904-0.076093270i
[2,] 0.02697303+0.009018432i -0.198193617+0i 0.14538171+0.016147937i
[3,] -0.03387892-0.040662080i -0.011105857+0i -0.08440467+0.029711907i
[4,] 0.06219272-0.013120331i -0.078596597+0i 0.07943484-0.005236883i
[5,] 0.04166513-0.023707111i -0.052668389+0i 0.02506748+0.004312003i
[6,] -0.01027128-0.013334092i 0.004203928+0i -0.02465543+0.005995309i
[,25] [,26]
[1,] -0.10373904+0.076093270i -0.008625179+0.004669923i
[2,] 0.14538171-0.016147937i -0.048809011+0.027032077i
[3,] -0.08440467-0.029711907i 0.038476787+0.036932686i
[4,] 0.07943484+0.005236883i -0.072430030+0.013290496i
[5,] 0.02506748-0.004312003i -0.052260860+0.018958707i
[6,] -0.02465543-0.005995309i 0.006002152+0.006952068i
[,27] [,28] [,29]
[1,] -0.008625179-0.004669923i -0.003718878+0i -0.009707559+0i
[2,] -0.048809011-0.027032077i 0.044557100+0i -0.022548972+0i
[3,] 0.038476787-0.036932686i -0.052466952+0i 0.019618559+0i
[4,] -0.072430030-0.013290496i 0.047300637+0i 0.037313074+0i
[5,] -0.052260860-0.018958707i 0.021593177+0i 0.015451784+0i
[6,] 0.006002152-0.006952068i -0.016074113+0i -0.004413701+0i
[,30] [,31]
[1,] -0.007550781+0.004600044i -0.007550781-0.004600044i
[2,] 0.031531413+0.010900162i 0.031531413-0.010900162i
[3,] 0.077445842+0.048091357i 0.077445842-0.048091357i
[4,] -0.001520130+0.056952785i -0.001520130-0.056952785i
[5,] 0.025512958+0.056900718i 0.025512958-0.056900718i
[6,] 0.014572071+0.017784406i 0.014572071-0.017784406i
[,32] [,33] [,34]
[1,] 0.004825424+0.004539263i 0.004825424-0.004539263i 0.004989896+0i
[2,] 0.007903530+0.057741523i 0.007903530-0.057741523i 0.007346722+0i
[3,] -0.066351587-0.093039152i -0.066351587+0.093039152i -0.018026633+0i
[4,] 0.017752755+0.094123085i 0.017752755-0.094123085i -0.005598798+0i
[5,] -0.062954327+0.023262635i -0.062954327-0.023262635i 0.009245646+0i
[6,] -0.003149832-0.035217586i -0.003149832+0.035217586i -0.003178692+0i
[,35] [,36] [,37]
[1,] -0.007773600+0i -0.0009083177+0i 0.002987234+0.0011800826i
[2,] -0.007073169+0i -0.0119645395+0i -0.002645267-0.0097695913i
[3,] 0.189512936+0i 0.0574572601+0i 0.187651643+0.1130183091i
[4,] -0.002264930+0i -0.0101011946+0i -0.026656976-0.0106903450i
[5,] 0.042318123+0i -0.0038313783+0i 0.019636549-0.0008084996i
[6,] 0.069051540+0i -0.5332661249+0i 0.114989113+0.0703393329i
[,38] [,39] [,40]
[1,] 0.002987234-0.0011800826i 0.0006681198+0i -0.004395855-6.758877e-05i
[2,] -0.002645267+0.0097695913i 0.0076396428+0i -0.020035269+2.806511e-04i
[3,] 0.187651643-0.1130183091i 0.0526193250+0i 0.182989060-1.974946e-02i
[4,] -0.026656976+0.0106903450i -0.0029212527+0i -0.029684364+8.753233e-04i
[5,] 0.019636549+0.0008084996i 0.0204250157+0i 0.010396574+3.289603e-03i
[6,] 0.114989113-0.0703393329i 0.1765315198+0i 0.047041038-1.145265e-03i
[,41] [,42] [,43]
[1,] -0.004395855+6.758877e-05i 0.001048224+0i -0.0004873891+0i
[2,] -0.020035269-2.806511e-04i -0.003388767+0i 0.0018716979+0i
[3,] 0.182989060+1.974946e-02i 0.409073599+0i 0.0140907317+0i
[4,] -0.029684364-8.753233e-04i -0.029677783+0i -0.0037872917+0i
[5,] 0.010396574-3.289603e-03i 0.061268855+0i -0.0013652710+0i
[6,] 0.047041038+1.145265e-03i 0.201553544+0i -0.0282238059+0i
[,44] [,45]
[1,] 8.610835e-04+0i 0.0011219722+0i
[2,] 5.848563e-05+0i 0.0021447645+0i
[3,] -3.141103e-02+0i 0.0059338088+0i
[4,] 6.262198e-03+0i 0.0006173083+0i
[5,] 7.121051e-04+0i -0.0001452607+0i
[6,] 2.372215e-02+0i -0.0052931041+0i> #Menyusun dimensi yang signifikan (misal, 2 dimensi)
> k <- 2
> eigenvaluess <- eigenvalues[1:k]
> eigenvaluess
[1] -7400232981+ 0i 1007811962+3812613341i
> eigenvectorss <- eigenvectors[, 1:k]
> eigenvectorss
[,1] [,2]
[1,] 0.040369428+0i 0.012837615+0.0203245607i
[2,] 0.054234055+0i -0.024664810+0.0609574427i
[3,] 0.038899444+0i -0.016543040+0.0441162206i
[4,] 0.071226961+0i 0.052944232+0.0160729994i
[5,] 0.067601593+0i -0.037404977+0.0386263862i
[6,] 0.012559384+0i -0.006282389+0.0064370160i
[7,] 0.151082305+0i -0.055329062-0.0106730698i
[8,] 0.009497537+0i -0.004686153+0.0100944464i
[9,] 0.029268785+0i -0.013536805+0.0178774238i
[10,] 0.037204331+0i -0.012590677-0.0087947908i
[11,] 0.030995245+0i -0.012786021+0.0192974866i
[12,] 0.030409765+0i -0.012721242+0.0270899041i
[13,] 0.208280773+0i -0.127400095+0.2546239308i
[14,] 0.032906338+0i -0.008844488+0.0165129263i
[15,] 0.410831133+0i -0.251679392+0.1359745929i
[16,] 0.022773150+0i -0.001191568+0.0090287592i
[17,] 0.072443715+0i -0.053773429-0.0031204787i
[18,] 0.009778560+0i 0.006518161+0.0018306609i
[19,] 0.012828573+0i -0.008868207+0.0068858529i
[20,] 0.019687733+0i -0.002414729+0.0122687633i
[21,] 0.008483625+0i 0.002915360+0.0027828396i
[22,] 0.178626666+0i -0.094057807+0.2659493618i
[23,] 0.020231489+0i 0.007407447+0.0072748763i
[24,] 0.020007738+0i -0.007071209+0.0029308237i
[25,] 0.066617620+0i 0.018687116+0.0311865470i
[26,] 0.016203343+0i 0.004370853+0.0021151340i
[27,] 0.051949312+0i 0.045893525+0.0026163588i
[28,] 0.089315548+0i 0.087897774+0.0146636270i
[29,] 0.177687976+0i 0.195934009+0.0101460099i
[30,] 0.052479816+0i 0.050471575+0.0089962886i
[31,] 0.125313467+0i 0.149356866+0.0039926526i
[32,] 0.117763642+0i 0.131351275+0.0107895567i
[33,] 0.134311201+0i -0.035930485+0.0675118050i
[34,] 0.103015793+0i 0.103891273-0.0067723135i
[35,] 0.055521937+0i 0.041766807+0.0138881958i
[36,] 0.033214430+0i -0.003977247+0.0316061514i
[37,] 0.119766405+0i -0.003685353+0.0101706709i
[38,] 0.088718774+0i 0.096056089+0.0014808172i
[39,] 0.062778410+0i 0.026088195+0.0021308392i
[40,] 0.069357352+0i 0.070922236+0.0001218985i
[41,] 0.120262739+0i 0.097209876-0.0214146947i
[42,] 0.634538140+0i 0.708581803+0.0000000000i
[43,] 0.130353594+0i 0.032535830-0.0031218253i
[44,] 0.247869066+0i 0.157529153-0.0814966856i
[45,] 0.256925046+0i -0.070367951-0.2422161455i> #Koordinat ruang 2 dimensi
> coordinates <- eigenvectors %*% diag(sqrt(eigenvalues))
> head(coordinates)
[,1] [,2] [,3] [,4]
[1,] 0+3472.764i -139.9403+1503.12072i -139.9403-1503.12072i 399.1834+0i
[2,] 0+4665.463i -3562.6845+2088.03643i -3562.6845-2088.03643i -4033.8726+0i
[3,] 0+3346.309i -2513.3397+1561.24804i -2513.3397-1561.24804i -3104.5421+0i
[4,] 0+6127.271i 2018.5058+2828.17869i 2018.5058-2828.17869i 2780.5135+0i
[5,] 0+5815.400i -3341.0209+ 488.81344i -3341.0209- 488.81344i -2740.9293+0i
[6,] 0+1080.416i -559.2090+ 79.58563i -559.2090- 79.58563i -575.5959+0i
[,5] [,6] [,7]
[1,] 0+ 644.8825i -1832.5104- 157.0719i -1832.5104+ 157.0719i
[2,] 0+ 2840.1525i -4546.0835-1244.5775i -4546.0835+1244.5775i
[3,] 0+ 581.9309i -4410.1099-1728.2243i -4410.1099+1728.2243i
[4,] 0+ 1817.1739i 838.7139+2722.6220i 838.7139-2722.6220i
[5,] 0-33033.9050i 118.7653-1412.5644i 118.7653+1412.5644i
[6,] 0- 514.7909i -307.7300- 761.8801i -307.7300+ 761.8801i
[,8] [,9] [,10]
[1,] 725.92819+ 230.5868i 725.92819- 230.5868i -379.4850+ 476.7718i
[2,] 1654.69599+2653.5998i 1654.69599-2653.5998i -1878.5021- 118.1139i
[3,] 1784.72909+2107.3696i 1784.72909-2107.3696i -1251.7100+ 580.8659i
[4,] 1267.03794+ 954.9051i 1267.03794- 954.9051i -417.1393+ 215.4195i
[5,] -27.84102- 263.8920i -27.84102+ 263.8920i 1628.6304+1087.7074i
[6,] -91.54621+ 789.8320i -91.54621- 789.8320i -161.3871- 585.1741i
[,11] [,12] [,13]
[1,] -379.4850- 476.7718i 319.76243+ 540.6656i 319.76243- 540.6656i
[2,] -1878.5021+ 118.1139i -1478.65175+2570.8123i -1478.65175-2570.8123i
[3,] -1251.7100- 580.8659i -364.92785+ 684.9752i -364.92785- 684.9752i
[4,] -417.1393- 215.4195i -674.49251+2014.8632i -674.49251-2014.8632i
[5,] 1628.6304-1087.7074i 40.71523+ 660.4082i 40.71523- 660.4082i
[6,] -161.3871+ 585.1741i 511.65567- 392.0812i 511.65567+ 392.0812i
[,14] [,15] [,16]
[1,] -351.5938-266.24977i -351.5938+266.24977i 345.9357+ 517.0498i
[2,] -1064.3577- 21.50041i -1064.3577+ 21.50041i 1839.4622+1438.0649i
[3,] -528.3240-291.28364i -528.3240+291.28364i 725.8162+ 226.8255i
[4,] 400.7461+114.13028i 400.7461-114.13028i 378.4727+ 992.4527i
[5,] -744.7023-100.09689i -744.7023+100.09689i 528.1093+ 966.9719i
[6,] -202.1837-222.22226i -202.1837+222.22226i 261.4105+ 160.7274i
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[1,] 345.9357- 517.0498i 508.6773+0i 328.070022+374.13499i
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[3,] 725.8162- 226.8255i 556.4125+0i 64.944097+107.69332i
[4,] 378.4727- 992.4527i -111.4357+0i 580.987602+664.46533i
[5,] 528.1093- 966.9719i 577.0320+0i 451.348802+540.65758i
[6,] 261.4105- 160.7274i 132.3816+0i 3.047698- 34.79013i
[,20] [,21] [,22]
[1,] 328.070022-374.13499i 134.07448- 64.81277i 134.07448+ 64.81277i
[2,] 1095.847193-808.32773i 167.12749+ 39.15057i 167.12749- 39.15057i
[3,] 64.944097-107.69332i -302.87519+101.50557i -302.87519-101.50557i
[4,] 580.987602-664.46533i 277.88098+264.47523i 277.88098-264.47523i
[5,] 451.348802-540.65758i 138.89110+253.80547i 138.89110-253.80547i
[6,] 3.047698+ 34.79013i -95.01346+ 35.94310i -95.01346- 35.94310i
[,23] [,24] [,25]
[1,] -572.259516+0i -150.48975-131.393413i -150.48975+131.393413i
[2,] -398.028838+0i 222.81370+ 44.126827i 222.81370- 44.126827i
[3,] -22.303701+0i -134.50610+ 34.857930i -134.50610- 34.857930i
[4,] -157.844197+0i 123.59401+ 2.356348i 123.59401- 2.356348i
[5,] -105.773021+0i 38.21756+ 9.972279i 38.21756- 9.972279i
[6,] 8.442677+0i -38.93718+ 6.029823i -38.93718- 6.029823i
[,26] [,27] [,28] [,29]
[1,] -9.204663- 6.344185i -9.204663+ 6.344185i -3.791783+0i -8.616865+0i
[2,] -52.703170-35.587899i -52.703170+35.587899i 45.430597+0i -20.015479+0i
[3,] -17.605715+58.183080i -17.605715-58.183080i -53.495513+0i 17.414313+0i
[4,] -50.966008-66.686424i -50.966008+66.686424i 48.227918+0i 33.120758+0i
[5,] -46.289307-43.270009i -46.289307+43.270009i 22.016490+0i 13.715696+0i
[6,] -3.955777+ 9.692209i -3.955777- 9.692209i -16.389230+0i -3.917799+0i
[,30] [,31] [,32]
[1,] -7.6468725+ 0.8522073i -7.6468725- 0.8522073i 3.2193738+ 3.823467i
[2,] 20.8628330+20.1900692i 20.8628330-20.1900692i 0.8758290+43.961899i
[3,] 43.4042625+66.4050603i 43.4042625-66.4050603i -41.5911120-75.522459i
[4,] -22.1378279+44.3624520i -22.1378279-44.3624520i 5.0767594+72.086868i
[5,] -0.7964213+54.2601646i -0.7964213-54.2601646i -49.2108312+11.930193i
[6,] 4.9551953+19.3848519i 4.9551953-19.3848519i 0.7178057-26.667119i
[,33] [,34] [,35] [,36]
[1,] 3.2193738- 3.823467i 3.316216+0i 0- 3.2025270i -0.339998+0i
[2,] 0.8758290-43.961899i 4.882530+0i 0- 2.9139673i -4.478521+0i
[3,] -41.5911120+75.522459i -11.980252+0i 0+78.0745474i 21.507183+0i
[4,] 5.0767594-72.086868i -3.720884+0i 0- 0.9330941i -3.781041+0i
[5,] -49.2108312-11.930193i 6.144529+0i 0+17.4339990i -1.434147+0i
[6,] 0.7178057+26.667119i -2.112515+0i 0+28.4474920i -199.610144+0i
[,37] [,38] [,39]
[1,] 1.0490759+ 0.5560844i 1.0490759- 0.5560844i 0.2337797+0i
[2,] -0.5711081- 3.6977622i -0.5711081+ 3.6977622i 2.6731634+0i
[3,] 64.3055618+49.2191359i 64.3055618-49.2191359i 18.4118628+0i
[4,] -9.3550161- 5.0209731i -9.3550161+ 5.0209731i -1.0221664+0i
[5,] 7.2474476+ 0.5084351i 7.2474476- 0.5084351i 7.1468531+0i
[6,] 39.3606763+30.5587413i 39.3606763-30.5587413i 61.7695899+0i
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[1,] -0.8353238- 0.9643782i -0.8353238+ 0.9643782i 0.2887508+0i
[2,] -3.9346110- 4.2815805i -3.9346110+ 4.2815805i -0.9334920+0i
[3,] 39.6554376+35.7821654i 39.6554376-35.7821654i 112.6861092+0i
[4,] -5.9289838- 6.2547656i -5.9289838+ 6.2547656i -8.1752376+0i
[5,] 1.2983016+ 2.8859831i 1.2983016- 2.8859831i 16.8775226+0i
[6,] 9.3433636+ 9.9587404i 9.3433636- 9.9587404i 55.5212674+0i
[,43] [,44] [,45]
[1,] -0.08150336+0i 0.135232676+0i 0.10593068+0i
[2,] 0.31299359+0i 0.009185135+0i 0.20249732+0i
[3,] 2.35631439+0i -4.933083630+0i 0.56023886+0i
[4,] -0.63332764+0i 0.983474577+0i 0.05828299+0i
[5,] -0.22830665+0i 0.111835698+0i -0.01371475+0i
[6,] -4.71970946+0i 3.725549446+0i -0.49974690+0i2.7 Koordinat Objek 2 Dimensi
Setelah matriks jarak ditransformasi, MDS akan menghasilkan koordinat untuk setiap objek (misalnya, kota) dalam ruang dua dimensi. Koordinat ini mewakili posisi relatif objek berdasarkan kedekatannya.
> coord_fit <- round(cmdscale(dist_matrix,k=2),2)
> coord_fit
[,1] [,2]
[1,] -88.99 -45.80
[2,] 19.39 -54.86
[3,] -42.60 -33.93
[4,] -27.65 -13.87
[5,] 38.20 -45.00
[6,] -28.41 -18.57
[7,] -38.02 -7.18
[8,] -24.34 -14.30
[9,] -53.40 -2.26
[10,] -37.45 -5.73
[11,] -8.51 -3.31
[12,] -13.46 -8.53
[13,] -26.64 1.52
[14,] -11.36 -5.64
[15,] -7.87 -2.78
[16,] -6.61 2.85
[17,] -22.03 4.25
[18,] -30.00 7.60
[19,] -6.97 -4.30
[20,] -86.25 23.52
[21,] -21.85 11.26
[22,] -20.23 5.76
[23,] -13.72 4.04
[24,] -9.09 7.48
[25,] -23.19 13.31
[26,] 19.04 0.40
[27,] -27.21 25.33
[28,] 25.93 1.83
[29,] 42.27 -1.99
[30,] 64.30 -11.75
[31,] 28.49 -0.49
[32,] 54.95 -5.04
[33,] 52.00 -4.63
[34,] -0.64 22.77
[35,] 46.35 -0.47
[36,] 21.62 11.57
[37,] -54.64 43.39
[38,] 21.93 14.03
[39,] 37.29 13.13
[40,] 11.80 20.44
[41,] 28.96 16.57
[42,] 37.34 16.88
[43,] 100.62 -2.57
[44,] 22.83 24.02
[45,] 57.81 1.072.8 Visualisasi
Hasil koordinat dua dimensi dari MDS kemudian divisualisasikan dalam bentuk plot untuk memudahkan pemahaman. Visualisasi ini menunjukkan hubungan antar objek, sehingga pola atau kelompok objek yang serupa dapat dilihat secara langsung.
> plot(coord_fit,pch=16,xlim=c(-35,55),ylim=c(-20,20),main="Plot 2 Dimensi",xlab="Dimensi 1",ylab="Dimensi 2",col="cyan")
> text.default(coord_fit,pos=4,labels=CostOfLiving$`Country`,cex=0.7)
Error in eval(expr, envir, enclos): object 'CostOfLiving' not found
> abline(h=0,col="red")
> abline(v=0,col="red")
2.9 Jarak Objek dari Koordinat
Jarak antar objek dalam plot dihitung kembali untuk memastikan apakah hasil MDS akurat menggambarkan hubungan yang ada di matriks jarak asli.
2.10 Evaluasi Model
Mengevaluasi sejauh mana model MDS berhasil. Hal ini dilakukan dengan memeriksa seberapa baik model dapat merepresentasikan data asli.
3 HASIL dan PEMBAHASAN
3.1 Statistika Deskriptif
> summary(Data_Cost_Of_Living)
Country Cost of Living Index Rent Index Groceries Index
Length:45 Min. : 46.40 Min. :11.2 Min. : 42.80
Class :character 1st Qu.: 52.30 1st Qu.:18.7 1st Qu.: 48.40
Mode :character Median : 61.10 Median :22.5 Median : 59.60
Mean : 60.98 Mean :26.2 Mean : 61.32
3rd Qu.: 64.80 3rd Qu.:33.4 3rd Qu.: 71.20
Max. :101.10 Max. :67.2 Max. :109.10
Restaurant Price Index Local Purchasing Power Index
Min. :26.00 Min. : 2.30
1st Qu.:44.70 1st Qu.: 68.30
Median :53.20 Median :102.40
Mean :55.42 Mean : 95.14
3rd Qu.:63.60 3rd Qu.:120.20
Max. :97.00 Max. :182.50 - Statistika desktiptif dari biaya hidup tanpa sewa menunjukkan bahwa biaya paling rendah berada pada angka 46.40, sedangkan untuk biaya paling besar yaitu 101.10
- Statistika desktiptif dari biaya sewa menunjukkan bahwa biaya paling rendah sebesar 11.2, sedangkan untuk biaya paling besar adalah 67.2
- Statistika desktiptif dari biaya bahan makanan menunjukkan bahwa biaya paling rendah berada pada angka 42.80, sedangkan untuk biaya paling besar yaitu 109.10
- Statistika desktiptif dari biaya makan di restoran menunjukkan bahwa biaya paling rendah sebesar 26.00, sedangkan untuk biaya paling besar adalah 97.00
- Statistika desktiptif dari daya beli lokal menunjukkan bahwa biaya paling rendah berada pada angka 2.30, sedangkan untuk biaya paling besar yaitu 182.50
3.2 Matriks Jarak
- Nilai terkecil dalam dataset adalah 4.54, menunjukkan bahwa jarak terpendek antar pasangan dalam data adalah 4.54.
- Nilai terbesar dalam dataset adalah 195.02, mencerminkan jarak terjauh di antara pasangan data.
3.3 Visualisasi
> plot(coord_fit,pch=16,xlim=c(-35,55),ylim=c(-20,20),main="Plot 2 Dimensi",xlab="Dimensi 1",ylab="Dimensi 2",col="cyan")
> text.default(coord_fit,pos=4,labels=CostOfLiving$`Country`,cex=0.7)
Error in eval(expr, envir, enclos): object 'CostOfLiving' not found
> abline(h=0,col="red")
> abline(v=0,col="red")
Dari plot 2 dimensi secara keseluruhan terdapat beberapa kelompok negara yang memiliki kemiripan karakteristik biaya hidup ditinjau dari biaya hidup tanpa sewa, biaya sewa, biaya bahan makanan, biaya makan di restoran dan kemampuan daya beli lokal. Hal tersebut digambarkan dari kedekatan titik atau posisi negara-negara yang ada.
- Titik yang berdekatan merepresentasikan negara-negara yang memiliki kesamaan lebih tinggi berdasarkan metrik jarak yang digunakan dalam analisis.
- Titik yang berjauhan menunjukkan bahwa negara-negara tersebut memiliki karakteristik yang lebih berbeda.
- Pembagian plot menjadi empat kuadran untuk mengelompokkan negara:
- Kuadran I (kanan atas): Negara-negara di sini memiliki karakteristik tertentu yang mirip berdasarkan dimensi positif di X dan Y. 2.Kuadran II (kiri atas): Negara-negara di sini memiliki karakteristik positif pada Y, tetapi negatif pada X. 3.Kuadran III (kiri bawah): Negara dengan nilai negatif pada X dan Y. 4.Kuadran IV (kanan bawah): Negara dengan nilai positif pada X, tetapi negatif pada Y.
- Nama-nama negara yang tumpang tindih bisa menunjukkan bahwa mereka sangat berdekatan dalam ruang dua dimensi
3.4 Jarak Objek dari Koordinat
Berdasarkan matriks tersebut menunjukkan bahwa jarak terdekat sebesar 0.83 dan jarak terjauh yang dihasilkan yaitu sebesar 194.48. Jarak terdekat menandakan kedua negara tersebut memiliki karakteristik biaya hidup yang mirip, sebaliknya jarak terjauh menandakan bahwa kedua negara memiliki karakteristik biaya hidup yang berbeda.
3.5 Evaluasi Model
Nilai STRESS yang diperoleh sebesar 0.0063, hal ini menunjukkan bahwa model penskalaan yang didapatkan termasuk kriteria sempurna.
Nilai R Square tang diperoleh dalam analisis Multidimensional Scaling menunjukkan bahwa model penskalaan sangat mewakilii data karena bernilai 0.9999 atau 99.99%.
4 PENUTUP
4.1 Kesimpulan
Berdasarkan plot 2 dimensi yang terbentuk, dari 45 negara yang dianalisis terdapat banyak pasangan negara yang tumpang tindih. Hal ini menunjukkan bahwa negara negara yang tumpah tindih memiliki karakteristik biaya hidup yang mirip ditinjau dari biaya hidup tanpa sewa, biaya sewa, biaya bahan makanan, biaya makan di restoran dan kemampuan daya beli lokal.
4.2 Saran
Banyak amatan dapat diperkecil untuk memudahkan pengambilan keputusan atau interpretasi dari plot dua dimensi yang terbentuk.
4.3 Dafar Pustaka
Myrios. (2024). Cost of Living Index by Country by Number 2024 [Dataset]. Kaggle. Retrieved from https://www.kaggle.com/datasets/myrios/cost-of-living-index-by-country-by-number-2024
Putri, D. S. (n.d.). Analisis Positioning dengan Menggunakan Multidimensional Scaling Nonmetrik. Universitas Mulawarman. Retrieved from https://fmipa.unmul.ac.id/files/docs/[11]%20Devy%20Sintya%20Putri_Edit.pdf
Claudius, H. 2023. Pertemuan Kelima : Multi-Dimensional Scaling [Slide Powerpoint].