Principal Component Analysis (PCA) dan Factor Analysis (FA) pada Nutrisi Makanan

Team Penyusun

Rivaldi Rizky Taufani (23031554010)
Iqbal Wahyufirmansyah (23031554106)

BAB 1 Pendahuluan

A. Latar Belakang

Nutrisi makanan adalah hal yang sangat penting bagi kehidupan semua orang. Nutrisi bermanfaat sebagai faktor yang mempengaruhi kesehatan dan pencegahan penyakit. Di dalam penelitian epidemiologi nutrisi, analisis pola makan telah digunakan untuk memahami hubungan antara asupan makanan dan kesehatan dengan menggunakan metode statistik Principal Component Analysis(PCA) dan Factor Analysis(FA). Dipilihnya metode PCA dan FA karena metode tersebut dapat mengidentifikasi pola yang kompleks dan mengurangi dimensi data nutrisi(Varraso et al.,2012).

PCA adalah salah satu metode yang digunakan dalam epidemiologi nutrisi untuk mengidentifikasi pola makan berdasarkan korelasi antara berbagai kelompok makanan. PCA bekerja dengan pengelompokan makanan dengan pola konsumsi yang serupa sehingga dapat memberikan gambaran yang lebih menyeluruh mengenai kebiasaan makan seseorang. Dalam jurnal Smith menunjukkan bahwa penggunaan PCA dalam analisis pola makan dapat dipengaruhi oleh cara kuantifikasi variabel input(berat makanan yang dikonsumsi, kontribusi energi, atau variabel biner)(Smith et al., 2013). Selain itu, metode FA lebih berfokus pada menemukan faktor laten yang mendasari pola konsumsi makanan dan memberikan pemahaman lebih dalam mengenai struktur pola diet yang tersembunyi (Varraso et al., 2012). Menurut sumber yang membandingkan PCA dan FA dalam analisis pola makan menunjukkan bahwa FA bisa lebih stabil dan dapat menghasilkan pola yang lebih mudah diinterpretasikan dalam beberapa kondisi dataset (Varraso et al., 2012). Namun, PCA tetap menjadi metode yang mendominasi karena kemampuannya dalam menangkap variasi utama dalam data nutrisi dengan pendekatan yang lebih eksploratif (Cozzolino et al., 2019).

Oleh karena itu, pentingnya mengidentifikasi pola makan yang dapat digunakan dalam intervensi kesehatan sehingga tujuan kami untuk membandingkan penggunaan PCA dan FA dalam menganalisis data nutrisi makanan serta mengevaluasi efektivitas masing-masing metode dalam mengekstraksi informasi utama dari dataset nutrisi.

B. Rumusan Masalah

Bagaimana penerapan PCA dan FA dalam analisis data nutrisi makanan?
Sejauh mana PCA dan FA dapat menyederhanakan variabel dalam dataset nutrisi tanpa kehilangan informasi yang signifikan?
Metode mana yang lebih efektif dalam mengidentifikasi faktor utama dalam nutrisi makanan?

C. Tujuan

Menganalisis pola utama dalam data nutrisi makanan menggunakan PCA dan FA.
Membandingkan efektivitas PCA dan FA dalam mengekstraksi informasi utama dari dataset nutrisi.
Memberikan rekomendasi terkait metode yang lebih optimal dalam penelitian nutrisi makanan.

BAB 2 Metodologi Penelitian

A. Data

Data yang digunakan dalam penelitian ini diperoleh dari dataset Fast Food Nutrition yang tersedia di Kaggle (https://www.kaggle.com/datasets/ulrikthygepedersen/fastfood-nutrition?resource=download). Dataset ini berisi informasi nutrisi dari berbagai produk makanan cepat saji yang dikumpulkan dari beberapa restoran ternama. Dataset ini sangat berguna dalam analisis pola konsumsi makanan dan dampaknya terhadap kesehatan masyarakat. Dataset ini mencakup berbagai informasi mengenai kandungan nutrisi dari makanan cepat saji, seperti jumlah kalori, total lemak, protein, karbohidrat, natrium, dan gula. Data ini memungkinkan analisis lebih lanjut menggunakan metode Principal Component Analysis (PCA) dan Factor Analysis (FA) untuk mengidentifikasi pola nutrisi utama dalam makanan cepat saji.

B. Variabel

Variabel yang digunakan dalam penelitian ini terdiri dari variabel nutrisi yang terdapat dalam dataset. Variabel-variabel utama yang dianalisis dalam penelitian ini meliputi:

Calories (Kalori) – Jumlah energi yang diperoleh dari satu porsi makanan.
Total Fat (g) (Lemak total) – Jumlah total lemak dalam satu porsi makanan (gram).
Saturated Fat (g) (Lemak jenuh) – Kandungan lemak jenuh dalam makanan (gram).
Trans Fat (g) (Lemak trans) – Jumlah lemak trans dalam makanan (gram).
Cholesterol (mg) (Kolesterol) – Kandungan kolesterol dalam satu porsi makanan (miligram).
Sodium (mg) (Natrium) – Kandungan natrium dalam makanan (miligram).
Carbohydrates (g) (Karbohidrat) – Total karbohidrat dalam makanan (gram).
Sugars (g) (Gula) – Kandungan gula dalam makanan (gram).
Protein (g) (Protein) – Kandungan protein dalam makanan (gram).
Dietary Fiber (g) (Serat pangan) – Kandungan serat pangan dalam makanan (gram).
Variabel-variabel ini akan digunakan sebagai input dalam analisis PCA dan FA untuk mengidentifikasi pola utama dalam nutrisi makanan cepat saji serta mengevaluasi bagaimana faktor-faktor tertentu saling berkorelasi dalam komposisi nutrisi makana

Bab 3 Pembahasan

A. Statistika Deskriptif

Dari hasil tabel statistika deskriptif di atas, dapat dilihat bahwa beberapa variabel(calories, sodium, cal_fat) memiliki rentang nilai sangat besar, berarti terdapat makanan yang memiliki nutrisi lebih tinggi dari yang lain. Variabel trans_fat memiliki sangat banyak nilai 0 sampai rentang 50% sehingga kemungkinan bisa mendistorsi hubungan antar variabel nantinya.

Variabel-variabel dataset fastfood didominasi oleh distribusi yang tidak seimbang (skewed) sehingga nanti bisa mempengaruhi hasil dari PCA dan FA. Selain itu, variabel yang distribusinya lebih merata adalah variabel “total_carb” tetapi masih memiliki variasi besar antar sample data. Sedangkan variabel “trans_fat” memiliki nilai 0 sangat banyak sehingga nanti dapat mempengaruhi hasil dari analisis selanjutnya.

B. Asumsi

Sebelum masuk ke tahapan PCA dan FA. Dilakukan tahapan asumsi untuk menguji apakah dataset yang digunakan cocok untuk dilakukan PCA dan FA atau tidak. Asumsi Yang digunakan pada dataset ini ada 2, yakni asumsi Kaiser Meyer Olkin (KMO) Test dan Bartlett’s Test of Sphericity. Dari dua metode tersebut didapat hasil untuk KMO overall MSA nya = 0.82 dan untuk Bartlett’s Test of Sphericity nilai (𝒳²) = 9264.87, p value = 0. Yang jika dilihat dari nilai MSA nya maka variabel telah cukup baik untuk dilakukan PCA karena berada di rentang 0.8 ≦ KMO < 0.9. Kemudian untuk nilai (𝒳²) = 9264.87 menunjukkan bahwa terdapat korelasi yang kuat antar variabelnya karena semakin besar 𝒳² maka semakin besar pula korelasi antar variabelnya. Serta p value < 0.05 yang berarti terdapat korelasi yang sangat kuat antar variabelnya. Dari hasil tersebut dapat disimpulkan bahwa dataset telah memenuhi asumsi yang digunakan sehingga dataset layak untuk diproses lebih lanjut untuk dilakukan PCA dan FA.

C. PCA

Pada diagram scree plot diatas menunjukkan seberapa besar persentase varians yang dijelaskan oleh komponen utama. Dapat dilihat pada diagaram di atas menunjukkan bahwa PC 1 mampu menjelaskan lebih banyak daripada PC yang lainnya. Kemudian nilai persentase varians menurun secara drastis dari sekitar 40% menjadi 18% untuk PC 2. Hal ini menunjukkan bahwa PC 1 merupakan komponen yang paling banyak dalam menjelaskan varians. Lalu untk PC 2 dan PC 3 hampir sama dalam menjelaskan varians karena persentasenya hampir mirip.

Gambar diatas menunjukkan 2 buah komponen PCA, yakni PC 1 dan PC 2. Yang mana berfungsi untuk menjelaskan sebagian besar dari varians yang ada pada dataset. Garis panah menunjukkan korelasi positif serta negatif antar variabel. Apabila searah maka menunjukkan variabel tersebut berkorelasi positif serta bila berbeda arah atau saling bertolak belakang maka menunjukkan korelasi negatif antar variabel. Serta titik biru menunjukkan observasi. Maka dengan biplot dapat dilakukan klasifikasi atau mengelompokkan data berdasarkan kemiripan antar variabelnya.

Plot ini berfungsi untuk melihat seberapa berkontribusi variabel terhadap PC nya. semakin merah warnanya maka variabel tersebut semakin berkontribusi dan semakin hijau warnanya maka variabel itu semakin tidak berkontribusi. Dari gambar diatas dapat dilihat ternyata terdapat beberapa variabel yang memiliki nilai kontribusi yang cukup tinggi dan ada juga yang memiliki kontribusi rendah terhadap PC.

Gambar di atas adalah gambar barplot untuk tiap variabel dalam PC 1. Yang mana garis merah putus-putus menandakan batas variabel yang masuk ke PC 1. Dapat dilihat bahwa data yang ada di PC 1 berhubungan dengan energi dan lemak. Yang mana variabel yang termasuk ke dalam PC 1 meliputi Calories, Cal_fat, Total_fat , Sat_fat, Protein, Cholesterol, Sodium.

Dari hasil PC2, menyatakan bahwa PC2 lebih ke dimensi nutrisi yang berkaitan dengan nutrisi penting dan serat. Jika PC1 menjelaskan variasi dalam hal energi dan lemak, maka PC2 lebih menjelaskan variasi dalam hal kesehatan dan keseimbangan nutrisi. PC2 membantu membedakan makanan berdasarkan kandungan vitamin dan serat yang tidak terwakili pada PC1. Dengan demikian, PC2 memberikan dimensi tambahan dalam hal kualitas nutrisi dari makanan yang dianalisis. Hal tersebut dapat berguna dalam klasifikasi makanan berdasarkan manfaat kesehatannya misalnya untuk membedakan antara makanan yang kaya energi dan lemak (PC1) dengan makanan yang kaya vitamin, mineral, dan serat (PC2).

PC3 didominasi oleh vit_a, total_carb, vit_c trans_fat. Vitamin A memiliki kontribusi terbesar dalam PC3 artinya dimensi ini lebih menekankan variasi dalam kandungan vitamin dan karbohidrat dalam dataset.

Secara keseluruhan, PCA menunjukkan bahwa variasi dalam dataset dapat dijelaskan melalui tiga aspek utama yaitu pertama kandungan energi dan lemak (PC1), kedua yaitu mikronutrien dan serat (PC2), dan ketiga vitamin serta karbohidrat (PC3). Hal ini dapat membantu dalam interpretasi lebih lanjut mengenai struktur data dan klasifikasi jenis makanan berdasarkan kandungan nutrisinya.

D. FA

Hasil dari FA menyatakan ada 3 kelompok variabel yang menjadi faktor utama dalam pola nutrisi makanan cepat saji. Pada MR1 memiliki variabel dengan loading tinggi yaitu total_fat, sat_fat, cal_fat, cholesterol, calories, dan trans_fat. Variabel ini berarti memiliki hubungan dengan kandungan lemak dan energi dalam makanan. Pada MR2 berisi variabel sodium, al_carb, fiber, dan sugar. Variabel ini berarti memiliki hubungan dengan karbohidrat dan kadar natrium dalam makanan cepat saji. Pada MR3 berisi variabel calcium, vit_c, dan vit_a. Variabel ini berarti memiliki hubungan dengan nutrisi penting yang dibutuhkan oleh tubuh dalam makanan.

Hasilnya FA menunjukkan 3 dimensi utama yaitu pertama adalah kandungan lemak dan energi, kedua adalah kandungan karbohidrat dan natrium, dan ketiga adalah nutrisi penting dalam makanan cepat saji. Hasil FA konsisten dengan uji asumsi yang telah dilakukan sebelumnya di mana nilai KMO sebesar 0.82 berarti dataset layak untuk dilakukan analisis faktor. Sementara hasil Bartlett’s Test menyatakan adanya korelasi kuat antar variabel yang mendukung penerapan metode FA. Selain itu, temuan ini sejalan dengan penelitian sumber referensi sebelumnya yang menunjukkan bahwa FA dapat mengidentifikasi struktur pola dengan lebih jelas dibandingkan PCA dalam beberapa kondisi dataset.

Bab 4 Kesimpulan

Dari eksperimen yang dilakukan dapat terhadap dataset nutrisi dalam makanan dapat disimpulkan. Bahwa metode Principal Component Analysis (PCA) dan Factor Analysis (FA) dapat digunakan untuk menganalisis pola nutrisi makanan cepat saji dari dataset Fast Food Nutrition. Dengan hasil PCA menunjukkan bahwa variasi dalam data dapat dijelaskan melalui tiga komponen utama yaitu PC1 (energi dan lemak), PC2 (mikronutrien dan serat), serta PC3 (vitamin dan karbohidrat). Sementara itu, FA mengidentifikasi tiga faktor utama dalam pola nutrisi, yaitu kandungan lemak dan energi, kandungan karbohidrat dan natrium, serta nutrisi penting (vitamin dan mineral). Uji asumsi (KMO = 0.82, Bartlett’s Test signifikan) menunjukkan bahwa dataset layak untuk PCA dan FA. Hasil FA lebih stabil dalam mengelompokkan faktor nutrisi yang mendasari sedangkan PCA lebih eksploratif dalam menangkap variasi utama. Secara keseluruhan, kedua metode memberikan informasi penting tentang struktur nutrisi makanan cepat saji yang dimana PCA lebih unggul dalam mengurangi dimensi data, sedangkan FA lebih efektif dalam menemukan faktor laten yang mendasari pola konsumsi makanan.

Daftar Pustaka

Smith, A. D. A. C., Emmett, P. M., Newby, P. K., & Northstone, K. (2013). Dietary patterns obtained through principal components analysis: The effect of input variable quantification. British Journal of Nutrition, 109(10), 1881-1891. https://doi.org/10.1017/S0007114512003868

Varraso, R., Garcia-Aymerich, J., Monier, F., Le Moual, N., De Batlle, J., Miranda, G., et al. (2012). Assessment of dietary patterns in nutritional epidemiology: Principal component analysis compared with confirmatory factor analysis. American Journal of Clinical Nutrition, 96(5), 1079-1092. https://doi.org/10.3945/ajcn.112.038109

Cozzolino, D., Power, A., & Chapman, J. (2019). Interpreting and reporting principal component analysis in food science analysis and beyond. Food Analytical Methods, 12(10), 2469-2473. https://doi.org/10.1007/s12161-019-01605-5

Source Code

data <- read.csv("C:/Users/USER/Downloads/fastfood_clean.csv")
print(head(data))

##   calories cal_fat total_fat sat_fat trans_fat cholesterol sodium total_carb
## 1      380      60       7.0     2.0       0.0          95   1110         44
## 2      840     410      45.0    17.0       1.5         130   1580         62
## 3     1130     595      66.5    21.5       2.5         185   1920         63
## 4      750     280      31.0    10.0       0.5         155   1940         62
## 5      920     410      45.0    12.0       0.5         120   1980         81
## 6      540     250      28.0    10.0       1.0          80    950         46
##   fiber sugar protein vit_a vit_c calcium
## 1     3    11      37     4    20      20
## 2     2    18      46     6    20      20
## 3     3    18      66    10    20      40
## 4     2    18      55     6    25      20
## 5     4    18      46     6    20      20
## 6     3     9      25    10     2      15

data_numeric <- data[, sapply(data, is.numeric)]

if (ncol(data_numeric) == 0) {
  stop("Tidak ada kolom numerik dalam dataset!")
}

library(psych)

## Warning: package 'psych' was built under R version 4.4.3

Melakukan Uji KMO (Kaiser-Meyer-Olkin)

r <- cor(data_numeric, use = "complete.obs")  
kmo_result <- KMO(r)
print(kmo_result)

## Kaiser-Meyer-Olkin factor adequacy
## Call: KMO(r = r)
## Overall MSA =  0.82
## MSA for each item = 
##    calories     cal_fat   total_fat     sat_fat   trans_fat cholesterol 
##        0.90        0.83        0.85        0.83        0.82        0.81 
##      sodium  total_carb       fiber       sugar     protein       vit_a 
##        0.94        0.73        0.78        0.83        0.84        0.57 
##       vit_c     calcium 
##        0.59        0.79

Uji Bartlett

bartlett_result <- cortest.bartlett(r, n = nrow(data_numeric))
print(bartlett_result)

## $chisq
## [1] 9264.87
## 
## $p.value
## [1] 0
## 
## $df
## [1] 91

Matriks Covarians

data_numeric <- data[, sapply(data, is.numeric)]

scale_data <- scale(data_numeric)

cov_matrix <- cov(scale_data, use = "complete.obs")  

print(cov_matrix)

##                 calories     cal_fat   total_fat      sat_fat    trans_fat
## calories     1.000000000  0.89191115  0.89101030  0.769175237  0.529215782
## cal_fat      0.891911150  1.00000000  0.99442069  0.862759337  0.619157458
## total_fat    0.891010300  0.99442069  1.00000000  0.858098667  0.619126343
## sat_fat      0.769175237  0.86275934  0.85809867  1.000000000  0.776656626
## trans_fat    0.529215782  0.61915746  0.61912634  0.776656626  1.000000000
## cholesterol  0.776227696  0.78225204  0.77977553  0.780269040  0.592646560
## sodium       0.811581126  0.67376788  0.67255251  0.566119961  0.297258582
## total_carb   0.728659391  0.44653194  0.45055723  0.345841033  0.133574029
## fiber        0.301912519  0.05328017  0.05762854 -0.008986456 -0.130121706
## sugar        0.513623884  0.32398097  0.33099952  0.335886146  0.216551402
## protein      0.811384073  0.67127983  0.67014494  0.627850226  0.448420702
## vit_a       -0.034476696 -0.01841241 -0.01893699  0.093657133  0.004493232
## vit_c       -0.006297276 -0.13021509 -0.12774575 -0.094447912 -0.145420703
## calcium      0.340169028  0.19053508  0.19212069  0.311765564  0.135955660
##             cholesterol      sodium  total_carb        fiber     sugar
## calories     0.77622770  0.81158113  0.72865939  0.301912519 0.5136239
## cal_fat      0.78225204  0.67376788  0.44653194  0.053280171 0.3239810
## total_fat    0.77977553  0.67255251  0.45055723  0.057628544 0.3309995
## sat_fat      0.78026904  0.56611996  0.34584103 -0.008986456 0.3358861
## trans_fat    0.59264656  0.29725858  0.13357403 -0.130121706 0.2165514
## cholesterol  1.00000000  0.64495080  0.29588560 -0.053685719 0.4170543
## sodium       0.64495080  1.00000000  0.69677315  0.327310552 0.4761922
## total_carb   0.29588560  0.69677315  1.00000000  0.645030754 0.5937284
## fiber       -0.05368572  0.32731055  0.64503075  1.000000000 0.3088025
## sugar        0.41705427  0.47619218  0.59372838  0.308802544 1.0000000
## protein      0.88071708  0.75447987  0.51731729  0.177336423 0.5040590
## vit_a       -0.01251788 -0.04024604 -0.05572428  0.233089090 0.1048313
## vit_c        0.01209290  0.07142676  0.10920779  0.430740891 0.3223571
## calcium      0.14545419  0.26655517  0.46688431  0.446175941 0.3797514
##                  protein        vit_a        vit_c   calcium
## calories     0.811384073 -0.034476696 -0.006297276 0.3401690
## cal_fat      0.671279833 -0.018412410 -0.130215091 0.1905351
## total_fat    0.670144944 -0.018936993 -0.127745754 0.1921207
## sat_fat      0.627850226  0.093657133 -0.094447912 0.3117656
## trans_fat    0.448420702  0.004493232 -0.145420703 0.1359557
## cholesterol  0.880717084 -0.012517882  0.012092902 0.1454542
## sodium       0.754479868 -0.040246038  0.071426762 0.2665552
## total_carb   0.517317295 -0.055724281  0.109207790 0.4668843
## fiber        0.177336423  0.233089090  0.430740891 0.4461759
## sugar        0.504059038  0.104831312  0.322357150 0.3797514
## protein      1.000000000 -0.008500396  0.125649980 0.2786604
## vit_a       -0.008500396  1.000000000  0.646211945 0.3495084
## vit_c        0.125649980  0.646211945  1.000000000 0.3702878
## calcium      0.278660403  0.349508369  0.370287832 1.0000000

Eigen value

pc <- eigen(r)
print(pc$values)

##  [1] 6.816508193 2.604247876 1.435911259 0.820125615 0.664797468 0.467602025
##  [7] 0.399717867 0.260826187 0.215474735 0.145291385 0.094497476 0.043493563
## [13] 0.026143640 0.005362711

p <- ncol(scale_data)
library(dplyr)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

Menghitung cumvar

sumvar <- sum(pc$values)
propvar <- (pc$values / sumvar) * 100  
cumvar <- data.frame(cbind(pc$values, propvar)) %>% 
  mutate(cum = cumsum(propvar))  
colnames(cumvar)[1] <- "value"
rownames(cumvar) <- paste0("PC", 1:p)
print(cumvar)

##            value     propvar       cum
## PC1  6.816508193 48.68934424  48.68934
## PC2  2.604247876 18.60177054  67.29111
## PC3  1.435911259 10.25650899  77.54762
## PC4  0.820125615  5.85804010  83.40566
## PC5  0.664797468  4.74855334  88.15422
## PC6  0.467602025  3.34001447  91.49423
## PC7  0.399717867  2.85512762  94.34936
## PC8  0.260826187  1.86304420  96.21240
## PC9  0.215474735  1.53910525  97.75151
## PC10 0.145291385  1.03779561  98.78930
## PC11 0.094497476  0.67498197  99.46429
## PC12 0.043493563  0.31066831  99.77495
## PC13 0.026143640  0.18674028  99.96169
## PC14 0.005362711  0.03830508 100.00000

Hasil PCA

print(pc$vectors)

##             [,1]         [,2]        [,3]        [,4]         [,5]        [,6]
##  [1,] 0.36949057  0.001368424 -0.11779416  0.04143723  0.116329233  0.02441011
##  [2,] 0.34540359 -0.165402558  0.05993035  0.13014083  0.224890630  0.09088434
##  [3,] 0.34537867 -0.162511046  0.05731728  0.12974522  0.218931238  0.10031320
##  [4,] 0.32865253 -0.156014542  0.23810568  0.22973370 -0.069592999  0.04006214
##  [5,] 0.24054731 -0.231763871  0.29932287  0.25958106 -0.330015924  0.35916228
##  [6,] 0.32677213 -0.148937976  0.14838810 -0.34640634 -0.002603662 -0.21534372
##  [7,] 0.31543430  0.074924069 -0.22665125 -0.18386601  0.243385248 -0.13423113
##  [8,] 0.25373980  0.275119483 -0.42614275  0.16379995 -0.036899773  0.13612374
##  [9,] 0.09796561  0.469618235 -0.24899949  0.25854031  0.282715130  0.26025096
## [10,] 0.21938954  0.254035276 -0.08794738 -0.34124433 -0.636223548  0.38271456
## [11,] 0.32778691  0.011107884 -0.01992010 -0.38876491  0.006078372 -0.34414225
## [12,] 0.01439324  0.341003319  0.60405821  0.04474178  0.273260507  0.12267613
## [13,] 0.01990642  0.477681518  0.35792357 -0.33550559  0.107212497  0.08370616
## [14,] 0.15023891  0.367460061  0.12183502  0.46338251 -0.377745102 -0.64254751
##              [,7]        [,8]         [,9]       [,10]       [,11]       [,12]
##  [1,] -0.06366507 -0.08788376  0.046301085 -0.32884533 -0.05047819  0.11968035
##  [2,] -0.25953516 -0.29959462 -0.063269911 -0.01850633 -0.19693010 -0.07476747
##  [3,] -0.26171356 -0.30816262 -0.062734724 -0.02595621 -0.22993039 -0.06297561
##  [4,] -0.02376597  0.04546763 -0.007883432  0.37177870  0.69386350 -0.33625847
##  [5,]  0.53374224  0.24414120 -0.199441569 -0.17073871 -0.25875023  0.09704868
##  [6,]  0.17935472 -0.13063633  0.283935773  0.16078493  0.17896358  0.68936514
##  [7,] -0.06502865  0.55764149 -0.485842416  0.37946129 -0.18028262  0.06654421
##  [8,] -0.07568636  0.20953169 -0.002761977 -0.50588738  0.41603502  0.15284151
##  [9,]  0.44439392 -0.18572532  0.301922839  0.39525682 -0.11704566 -0.01420631
## [10,] -0.33803264 -0.05297244  0.132424587  0.22043391 -0.15211194 -0.06635545
## [11,]  0.32361011  0.08356730  0.298033096 -0.23649455 -0.12073453 -0.58052754
## [12,] -0.28337394  0.44056431  0.362712609 -0.10728263 -0.08203656  0.04968885
## [13,]  0.17496221 -0.35531213 -0.545312414 -0.15278152  0.18445580 -0.02561239
## [14,] -0.07166109 -0.09655347 -0.083456127  0.04591554 -0.16619525  0.07666342
##               [,13]        [,14]
##  [1,]  0.8317886085 -0.054317923
##  [2,] -0.1942224052  0.728298931
##  [3,] -0.2938436638 -0.681869084
##  [4,]  0.1240148051 -0.030286425
##  [5,] -0.0600152593  0.012314098
##  [6,] -0.1399889609  0.009323147
##  [7,]  0.0169994710 -0.003208779
##  [8,] -0.3593959354  0.020816478
##  [9,]  0.0093325414  0.002324827
## [10,]  0.0334594911  0.006394181
## [11,] -0.1198152795  0.004626427
## [12,] -0.0152567856  0.001111003
## [13,]  0.0009010252 -0.001589051
## [14,] -0.0152486082  0.003725770

scores <- as.matrix(scale_data) %*% pc$vectors
head(scores)

##            [,1]        [,2]        [,3]       [,4]       [,5]       [,6]
## [1,] -1.0840761  0.19211454 -1.14236133 -1.8247706 -1.2837802 -0.6300620
## [2,]  3.8849715 -1.10561722 -0.03622534 -1.0469920 -1.7364342  0.8703269
## [3,]  7.2219520 -1.07357037  1.05081753 -0.1570991 -1.8924639 -0.1825995
## [4,]  2.9518100 -0.08988460 -0.75553145 -2.6737912 -1.4048995 -0.2635186
## [5,]  3.7613023  0.04355498 -1.44982671 -1.3429175 -0.8032773  0.5877690
## [6,]  0.2624154 -1.57346363 -0.43991475  0.2685139 -0.8092863  0.7245602
##             [,7]       [,8]       [,9]       [,10]       [,11]       [,12]
## [1,]  0.66830854 -0.1845051  0.1405559 -0.02838024  0.06517394  0.25022586
## [2,] -0.11601127 -0.4949331 -0.3500265  0.01006325  0.38394233 -0.35027178
## [3,]  0.41097471 -0.7616522 -0.1419161 -0.09345591 -0.45848903 -0.30094747
## [4,] -0.02509981 -0.1719767 -0.1487021  0.02414976  0.23564258  0.08333498
## [5,] -0.68615765 -0.4843246 -0.2038503 -0.07186457  0.16748657 -0.13451978
## [6,]  0.04733827  0.1726671  0.6068230  0.07811249  0.05728373  0.09554407
##            [,13]         [,14]
## [1,] -0.08172227  0.0014248811
## [2,]  0.03204832  0.0261103468
## [3,] -0.04873867  0.0005182989
## [4,] -0.04153297  0.0129070947
## [5,]  0.02026708  0.0343449481
## [6,] -0.02086866 -0.0013030402

PCA.mod <- prcomp(scale_data)
summary(PCA.mod)

## Importance of components:
##                           PC1    PC2    PC3     PC4     PC5    PC6     PC7
## Standard deviation     2.6108 1.6138 1.1983 0.90561 0.81535 0.6838 0.63223
## Proportion of Variance 0.4869 0.1860 0.1026 0.05858 0.04749 0.0334 0.02855
## Cumulative Proportion  0.4869 0.6729 0.7755 0.83406 0.88154 0.9149 0.94349
##                            PC8     PC9    PC10    PC11    PC12    PC13    PC14
## Standard deviation     0.51071 0.46419 0.38117 0.30740 0.20855 0.16169 0.07323
## Proportion of Variance 0.01863 0.01539 0.01038 0.00675 0.00311 0.00187 0.00038
## Cumulative Proportion  0.96212 0.97752 0.98789 0.99464 0.99775 0.99962 1.00000

PCA.mod$rotation

##                    PC1          PC2         PC3         PC4          PC5
## calories    0.36949057 -0.001368424 -0.11779416  0.04143723 -0.116329233
## cal_fat     0.34540359  0.165402558  0.05993035  0.13014083 -0.224890630
## total_fat   0.34537867  0.162511046  0.05731728  0.12974522 -0.218931238
## sat_fat     0.32865253  0.156014542  0.23810568  0.22973370  0.069592999
## trans_fat   0.24054731  0.231763871  0.29932287  0.25958106  0.330015924
## cholesterol 0.32677213  0.148937976  0.14838810 -0.34640634  0.002603662
## sodium      0.31543430 -0.074924069 -0.22665125 -0.18386601 -0.243385248
## total_carb  0.25373980 -0.275119483 -0.42614275  0.16379995  0.036899773
## fiber       0.09796561 -0.469618235 -0.24899949  0.25854031 -0.282715130
## sugar       0.21938954 -0.254035276 -0.08794738 -0.34124433  0.636223548
## protein     0.32778691 -0.011107884 -0.01992010 -0.38876491 -0.006078372
## vit_a       0.01439324 -0.341003319  0.60405821  0.04474178 -0.273260507
## vit_c       0.01990642 -0.477681518  0.35792357 -0.33550559 -0.107212497
## calcium     0.15023891 -0.367460061  0.12183502  0.46338251  0.377745102
##                     PC6         PC7         PC8          PC9        PC10
## calories    -0.02441011 -0.06366507  0.08788376 -0.046301085 -0.32884533
## cal_fat     -0.09088434 -0.25953516  0.29959462  0.063269911 -0.01850633
## total_fat   -0.10031320 -0.26171356  0.30816262  0.062734724 -0.02595621
## sat_fat     -0.04006214 -0.02376597 -0.04546763  0.007883432  0.37177870
## trans_fat   -0.35916228  0.53374224 -0.24414120  0.199441569 -0.17073871
## cholesterol  0.21534372  0.17935472  0.13063633 -0.283935773  0.16078493
## sodium       0.13423113 -0.06502865 -0.55764149  0.485842416  0.37946129
## total_carb  -0.13612374 -0.07568636 -0.20953169  0.002761977 -0.50588738
## fiber       -0.26025096  0.44439392  0.18572532 -0.301922839  0.39525682
## sugar       -0.38271456 -0.33803264  0.05297244 -0.132424587  0.22043391
## protein      0.34414225  0.32361011 -0.08356730 -0.298033096 -0.23649455
## vit_a       -0.12267613 -0.28337394 -0.44056431 -0.362712609 -0.10728263
## vit_c       -0.08370616  0.17496221  0.35531213  0.545312414 -0.15278152
## calcium      0.64254751 -0.07166109  0.09655347  0.083456127  0.04591554
##                    PC11        PC12          PC13         PC14
## calories    -0.05047819 -0.11968035  0.8317886085  0.054317923
## cal_fat     -0.19693010  0.07476747 -0.1942224052 -0.728298931
## total_fat   -0.22993039  0.06297561 -0.2938436638  0.681869084
## sat_fat      0.69386350  0.33625847  0.1240148051  0.030286425
## trans_fat   -0.25875023 -0.09704868 -0.0600152593 -0.012314098
## cholesterol  0.17896358 -0.68936514 -0.1399889609 -0.009323147
## sodium      -0.18028262 -0.06654421  0.0169994710  0.003208779
## total_carb   0.41603502 -0.15284151 -0.3593959354 -0.020816478
## fiber       -0.11704566  0.01420631  0.0093325414 -0.002324827
## sugar       -0.15211194  0.06635545  0.0334594911 -0.006394181
## protein     -0.12073453  0.58052754 -0.1198152795 -0.004626427
## vit_a       -0.08203656 -0.04968885 -0.0152567856 -0.001111003
## vit_c        0.18445580  0.02561239  0.0009010252  0.001589051
## calcium     -0.16619525 -0.07666342 -0.0152486082 -0.003725770

head(PCA.mod$x)

##             PC1         PC2         PC3        PC4       PC5        PC6
## [1,] -1.0840761 -0.19211454 -1.14236133 -1.8247706 1.2837802  0.6300620
## [2,]  3.8849715  1.10561722 -0.03622534 -1.0469920 1.7364342 -0.8703269
## [3,]  7.2219520  1.07357037  1.05081753 -0.1570991 1.8924639  0.1825995
## [4,]  2.9518100  0.08988460 -0.75553145 -2.6737912 1.4048995  0.2635186
## [5,]  3.7613023 -0.04355498 -1.44982671 -1.3429175 0.8032773 -0.5877690
## [6,]  0.2624154  1.57346363 -0.43991475  0.2685139 0.8092863 -0.7245602
##              PC7        PC8        PC9        PC10        PC11        PC12
## [1,]  0.66830854  0.1845051 -0.1405559 -0.02838024  0.06517394 -0.25022586
## [2,] -0.11601127  0.4949331  0.3500265  0.01006325  0.38394233  0.35027178
## [3,]  0.41097471  0.7616522  0.1419161 -0.09345591 -0.45848903  0.30094747
## [4,] -0.02509981  0.1719767  0.1487021  0.02414976  0.23564258 -0.08333498
## [5,] -0.68615765  0.4843246  0.2038503 -0.07186457  0.16748657  0.13451978
## [6,]  0.04733827 -0.1726671 -0.6068230  0.07811249  0.05728373 -0.09554407
##             PC13          PC14
## [1,] -0.08172227 -0.0014248811
## [2,]  0.03204832 -0.0261103468
## [3,] -0.04873867 -0.0005182989
## [4,] -0.04153297 -0.0129070947
## [5,]  0.02026708 -0.0343449481
## [6,] -0.02086866  0.0013030402

library(FactoMineR)

## Warning: package 'FactoMineR' was built under R version 4.4.3

pca_result <- PCA(scale_data, 
                  scale.unit = TRUE, 
                  graph = FALSE, 
                  ncp=p)     
pca_result$eig

##          eigenvalue percentage of variance cumulative percentage of variance
## comp 1  6.816508193            48.68934424                          48.68934
## comp 2  2.604247876            18.60177054                          67.29111
## comp 3  1.435911259            10.25650899                          77.54762
## comp 4  0.820125615             5.85804010                          83.40566
## comp 5  0.664797468             4.74855334                          88.15422
## comp 6  0.467602025             3.34001447                          91.49423
## comp 7  0.399717867             2.85512762                          94.34936
## comp 8  0.260826187             1.86304420                          96.21240
## comp 9  0.215474735             1.53910525                          97.75151
## comp 10 0.145291385             1.03779561                          98.78930
## comp 11 0.094497476             0.67498197                          99.46429
## comp 12 0.043493563             0.31066831                          99.77495
## comp 13 0.026143640             0.18674028                          99.96169
## comp 14 0.005362711             0.03830508                         100.00000

pca_result$svd$V

##             [,1]         [,2]        [,3]        [,4]         [,5]        [,6]
##  [1,] 0.36949057  0.001368424 -0.11779416  0.04143723  0.116329233  0.02441011
##  [2,] 0.34540359 -0.165402558  0.05993035  0.13014083  0.224890630  0.09088434
##  [3,] 0.34537867 -0.162511046  0.05731728  0.12974522  0.218931238  0.10031320
##  [4,] 0.32865253 -0.156014542  0.23810568  0.22973370 -0.069592999  0.04006214
##  [5,] 0.24054731 -0.231763871  0.29932287  0.25958106 -0.330015924  0.35916228
##  [6,] 0.32677213 -0.148937976  0.14838810 -0.34640634 -0.002603662 -0.21534372
##  [7,] 0.31543430  0.074924069 -0.22665125 -0.18386601  0.243385248 -0.13423113
##  [8,] 0.25373980  0.275119483 -0.42614275  0.16379995 -0.036899773  0.13612374
##  [9,] 0.09796561  0.469618235 -0.24899949  0.25854031  0.282715130  0.26025096
## [10,] 0.21938954  0.254035276 -0.08794738 -0.34124433 -0.636223548  0.38271456
## [11,] 0.32778691  0.011107884 -0.01992010 -0.38876491  0.006078372 -0.34414225
## [12,] 0.01439324  0.341003319  0.60405821  0.04474178  0.273260507  0.12267613
## [13,] 0.01990642  0.477681518  0.35792357 -0.33550559  0.107212497  0.08370616
## [14,] 0.15023891  0.367460061  0.12183502  0.46338251 -0.377745102 -0.64254751
##              [,7]        [,8]         [,9]       [,10]       [,11]       [,12]
##  [1,] -0.06366507 -0.08788376 -0.046301085 -0.32884533  0.05047819  0.11968035
##  [2,] -0.25953516 -0.29959462  0.063269911 -0.01850633  0.19693010 -0.07476747
##  [3,] -0.26171356 -0.30816262  0.062734724 -0.02595621  0.22993039 -0.06297561
##  [4,] -0.02376597  0.04546763  0.007883432  0.37177870 -0.69386350 -0.33625847
##  [5,]  0.53374224  0.24414120  0.199441569 -0.17073871  0.25875023  0.09704868
##  [6,]  0.17935472 -0.13063633 -0.283935773  0.16078493 -0.17896358  0.68936514
##  [7,] -0.06502865  0.55764149  0.485842416  0.37946129  0.18028262  0.06654421
##  [8,] -0.07568636  0.20953169  0.002761977 -0.50588738 -0.41603502  0.15284151
##  [9,]  0.44439392 -0.18572532 -0.301922839  0.39525682  0.11704566 -0.01420631
## [10,] -0.33803264 -0.05297244 -0.132424587  0.22043391  0.15211194 -0.06635545
## [11,]  0.32361011  0.08356730 -0.298033096 -0.23649455  0.12073453 -0.58052754
## [12,] -0.28337394  0.44056431 -0.362712609 -0.10728263  0.08203656  0.04968885
## [13,]  0.17496221 -0.35531213  0.545312414 -0.15278152 -0.18445580 -0.02561239
## [14,] -0.07166109 -0.09655347  0.083456127  0.04591554  0.16619525  0.07666342
##               [,13]        [,14]
##  [1,] -0.8317886085 -0.054317923
##  [2,]  0.1942224052  0.728298931
##  [3,]  0.2938436638 -0.681869084
##  [4,] -0.1240148051 -0.030286425
##  [5,]  0.0600152593  0.012314098
##  [6,]  0.1399889609  0.009323147
##  [7,] -0.0169994710 -0.003208779
##  [8,]  0.3593959354  0.020816478
##  [9,] -0.0093325414  0.002324827
## [10,] -0.0334594911  0.006394181
## [11,]  0.1198152795  0.004626427
## [12,]  0.0152567856  0.001111003
## [13,] -0.0009010252 -0.001589051
## [14,]  0.0152486082  0.003725770

pca_result$ind['coord']

## $coord
##           Dim.1        Dim.2        Dim.3         Dim.4        Dim.5
## 1   -1.08513018  0.192301330 -1.143472036 -1.8265447618 -1.285028382
## 2    3.88874882 -1.106692202 -0.036260563 -1.0480099343 -1.738122509
## 3    7.22897382 -1.074614196  1.051839231 -0.1572518141 -1.894303915
## 4    2.95467998 -0.089971990 -0.756266042 -2.6763909100 -1.406265513
## 5    3.76495933  0.043597333 -1.451236362 -1.3442231648 -0.804058298
## 6    0.26267057 -1.574993494 -0.440342472  0.2687749892 -0.810073123
## 7   -2.28126842 -1.448375320 -0.599150933 -0.0378236449 -0.822262029
## 8   -1.05171160 -1.817499673 -2.049709321 -0.8021076504  0.213893380
## 9   -0.27075268 -1.394285833  0.700548338  0.0780575472 -0.503876307
## 10   4.10188691 -2.241122705  1.830194233  0.1193028764 -0.896158554
## 11  -2.11124881 -1.683009217 -1.464599435  0.2859938964 -0.556969769
## 12   1.58724684 -0.841941615  0.433998392  0.9485746731 -1.871357104
## 13   0.63395657  0.498852234 -0.278163738 -1.0311136856 -1.250320850
## 14   1.38352034  0.660650711 -0.993947699  0.3308142238 -0.663501861
## 15  -3.20376678 -1.752508498 -1.453884634 -0.4342092075 -0.671770722
## 16  -2.69117181 -1.221306193 -1.131851812 -0.5342828487 -0.536708165
## 17   1.85492814 -1.708500017  0.153005643 -0.3992248671 -0.819979176
## 18   1.51882504  0.706130938 -0.770874764 -2.1916284457 -1.229448444
## 19   2.41972375  0.847404016 -1.296511465 -0.0484747734 -1.342682165
## 20  -2.50046251 -1.873438706 -1.585223240 -0.3206455010 -0.272392057
## 21  -1.03181259 -1.896295846 -0.228603956 -0.0989007877 -0.821301774
## 22  -0.70205722 -1.714958402 -1.614404422 -1.0790884870 -0.962685816
## 23   0.94372363 -1.471289557  0.245227385  0.2287359098 -0.475630982
## 24   0.59590376  0.868805384 -0.708052634 -1.4402912982 -1.004866249
## 25   1.21633725  0.810746727 -1.549577371  0.0347335729 -0.519787608
## 26  -0.20667634 -0.131454595 -1.834205229 -0.3341976292 -0.843935563
## 27  -0.54818275 -0.101192943 -1.598367516 -0.4460873863 -1.189310975
## 28   0.94004565 -1.280575754  0.830459321  0.0310995595 -0.825557975
## 29   1.95510046  0.008854737  0.837917367  0.5637736196 -1.318581742
## 30   0.88834911  1.521647690 -0.042096791 -1.5123957163 -0.798359889
## 31   1.48252717  1.421338247 -0.568239452 -0.1749730795 -0.266501158
## 32   2.32505709 -1.380240372 -0.154497990 -0.6300912486 -1.302825947
## 33   1.86474869  1.068105706 -1.272307250 -2.5563047738 -1.414513006
## 34   2.73244957  0.823457736 -1.719394135 -0.6888874419 -1.465627821
## 35  -2.11110290 -3.200830183 -1.100556937 -0.9428556537  0.525642766
## 36  -0.87834047 -3.285875162 -1.241115039 -1.3948158213  0.770158598
## 37   1.90066068 -3.645859707 -1.265135473 -1.7005969892  1.397301897
## 38   5.22580846 -3.959191457 -1.479001595 -1.6639252521  1.758281063
## 39   5.50788529 -3.860513785 -1.444225074 -1.2750143364  1.925696487
## 40   6.75394434 -3.667504177 -1.452603345 -0.2551872557  1.210084944
## 41  -3.96323894 -2.766114945 -1.008966228 -0.3028741570  0.147902388
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## 461  0.23936932 -0.315523029 -0.796000227  0.9436925939  0.978990051
## 462 -2.26584190  1.985524308 -0.817723103  1.3007115764  0.698631118
## 463 -2.04671943  1.535636657 -0.469640512  0.5904889422  0.368280524
## 464 -2.12287114  1.562353144 -0.511952860  0.7184732050  0.398369630
## 465 -3.69952446  0.224594415  0.810647753  0.1638238146 -0.277919018
## 466 -3.78584763  0.174636596  0.957367863  0.6364346019 -0.006244386
## 467 -3.72468103  0.243674769  0.791720501  0.3339074364 -0.276289590
## 468 -3.44674125  0.330375573  0.752752832  0.5229130750 -0.093018926
## 469 -0.08614312  0.065508820  0.890059964  0.9585974582 -0.120623404
## 470 -0.12470149  0.055288595  0.852044827  0.9954351827  0.016235596
## 471 -0.17114449  0.001511372  0.870662580  1.0676738588  0.150918998
## 472 -0.13007682  0.054011811  0.855907198  0.9985684488  0.012088061
## 473  0.79291180 -0.451055078 -0.168373496  1.8205117468 -0.036665053
## 474 -2.28478501  0.632330557  0.472883032  0.3986854474 -0.294976513
## 475 -2.52472635  0.314914358  0.687384866 -0.0700501812 -0.628608609
## 476 -2.60087806  0.341630845  0.645072518  0.0579340816 -0.598519503
## 477  0.27094527 -0.099811519 -2.402334151  1.3498075226  1.952488923
## 478  0.66969750  1.682215686 -1.145505039  1.6015592844  1.504640499
## 479 -1.52244541  0.987925845 -0.034714287  1.1640477320  0.797380443
## 480 -2.73803241  1.067721381 -0.089645325  1.2155380738  0.566630476
## 481 -3.01909822 -1.891780192 -0.849859897  0.5458538387  0.257033994
## 482 -1.60961192  0.577019069  0.117116780  0.7489848350  0.439855274
## 483 -3.08540838  0.095571515  0.883019584  0.6253819679  0.223360010
## 484 -1.74965789  0.841077764 -0.083942001  0.9278405478  0.530698689
## 485 -1.59846228  0.364612332  0.235020784  0.6972526758  0.315049549
## 486 -0.12093041  1.704947416 -0.956725495  0.5138019155  1.240375639
## 487 -0.08886733  1.823703794 -1.053497437  0.6670870167  1.306401436
## 488 -0.87656801  2.105012008 -1.314368111  1.3581277763  1.163316111
## 489 -0.25397955 -0.211477869  0.969442804  1.0962756561  0.174809467
## 490 -3.31213612 -0.035904376  1.017765602  0.5941697083 -0.043179967
## 491  0.40187337 -0.301299997  0.976953198  0.6914213051  0.296524473
## 492 -0.02063394 -1.155867360 -1.293934462  0.6985888859  0.878550921
## 493 -1.23074723  0.260389914  0.692070620  1.1864753360  0.103564084
## 494 -2.73817419 -0.299780704  0.920281377  0.5078465776  0.255954097
## 495  1.32315562 -0.632191012 -0.604606584  2.0009824029  0.042691363
## 496 -0.60338191  1.587321741 -0.782124183  0.8468308283  0.369762292
## 497 -2.60567745  1.299613198  0.006970665  1.1158088834  0.648052937
## 498  0.70652562  0.969641470 -0.027098483  1.0299817941  0.286445900
## 499 -1.13490540  0.743573474 -0.145897446  0.7050817465  0.337626250
## 500 -0.17922036  0.389618655  0.458177833  1.5499908186  0.737590486
## 501 -1.96348732 -0.294160433  1.322007533  0.8205858093 -0.272488736
## 502 -0.61660375 -0.624601281  0.345165878  1.4143133548 -0.488773501
## 503 -0.62644098 -0.516119124  0.386997493  1.3521848411 -0.486849212
## 504 -0.27115132 -0.644731833  0.440332982  1.9579572814 -0.517060645
## 505 -3.40511585  0.021784988  0.933682058  0.2971753408 -0.001284850
## 506 -1.79187269  0.379443947  0.229319372  0.5418800834  0.363640731
## 507 -3.41350156  0.455773560  0.617870036  0.8674486927  0.231387724
## 508 -1.16333104 -0.792291850 -0.738669428  1.4238059117 -0.511550665
## 509  0.30399816 -0.264361949  0.931035729  0.8112449006  0.312843321
## 510  1.07557702  0.928310868 -1.740104621  1.5974513538  0.527362761
## 511  1.87505305  1.209214640 -1.339293414  1.4978364733  0.957388864
## 512  0.70652562  0.969641470 -0.027098483  1.0299817941  0.286445900
## 513  1.94803496  1.127231617 -0.420537985  1.4695402955  0.900792458
## 514  1.14780797  1.387493253 -0.758068137  0.3766478537  0.876590009
## 515  1.42027626  1.069873606 -0.351900959  1.0043367216  0.432367347
##             Dim.6        Dim.7        Dim.8        Dim.9       Dim.10
## 1   -0.6306746293  0.668958333 -0.184684509 -0.140692528 -0.028407832
## 2    0.8711731066 -0.116124064 -0.495414299  0.350366815  0.010073032
## 3   -0.1827770090  0.411374301 -0.762392752  0.142054066 -0.093546771
## 4   -0.2637748184 -0.025124219 -0.172143936  0.148846694  0.024173237
## 5    0.5883405293 -0.686824794 -0.484795503  0.204048489 -0.071934448
## 6    0.7252647319  0.047384299  0.172835020 -0.607413040  0.078188440
## 7    0.6752631170 -0.034028638  0.485238032 -0.473459669  0.056441188
## 8    0.5980037084  0.040573589 -0.339661100 -0.252161199 -0.202475549
## 9    0.4310558077  0.003909225  1.008996939 -0.761026122  0.216114977
## 10   0.5735413201  1.116242053  0.579264298 -1.160469673  0.242942831
## 11  -0.0645370012 -0.324164432 -0.363312383 -0.267256854 -0.084586588
## 12  -0.1119890030  0.287350231 -0.470306215 -0.308939236 -0.088972654
## 13  -1.5036649191  0.090904600 -0.503358837 -0.457933449  0.050587726
## 14  -0.6337451392 -0.572806961 -0.780476655 -0.375312317 -0.062254489
## 15   0.6519707219  0.046860821 -0.095022180 -0.374941485  0.053027103
## 16  -0.6913531086  0.625676453  0.028914739 -0.453654075 -0.312404898
## 17   0.7048929978  0.759671645 -0.381887941  0.183442012  0.242760231
## 18  -1.2937534848  0.306398158 -0.445161275  0.589127627  0.485039448
## 19  -0.7881101598 -0.094319811 -0.626792367  0.527968941  0.206246750
## 20   0.5937701232 -0.088363753 -0.198393472 -0.315069690 -0.215593666
## 21   0.6716513016  0.393296212  0.538853316 -0.515001473  0.082026028
## 22   0.6973340144 -0.549753004 -0.348772308 -0.694984577  0.226439480
## 23   0.8824973653  0.997913508 -0.508441277  0.028919654  0.012042162
## 24  -1.0225995959  0.311581679 -0.322895319  0.099959693  0.340493614
## 25  -0.1884050253 -0.386630614 -0.603667396 -0.216762348  0.159411129
## 26   0.1405057452 -0.295095043 -0.410884579 -0.284912034 -0.254611958
## 27   0.3414137109 -0.753803015 -0.297600900 -0.192677562 -0.287455917
## 28   0.9486429692  0.327595894  0.871052380 -0.868395858  0.214432874
## 29   0.6519935332 -0.019694430  0.040731236 -0.524502800 -0.226731901
## 30  -0.6557724359 -0.308585571  0.250265252 -0.478574930  0.015808736
## 31   0.0387356890 -1.018122171 -0.091194612 -0.419523805 -0.380201737
## 32   1.0737332701  0.408789222 -0.333435639  0.092476696  0.144555958
## 33  -0.4992555053  0.144177641 -0.446413055  0.352948156  0.457652368
## 34   0.2205291133 -0.401008880 -0.505082119  0.368370117  0.061804737
## 35  -0.5244107060  0.222449337 -0.198992358 -0.131594157 -0.149241040
## 36  -0.7643349500  0.165768795 -0.109024535 -0.177222953 -0.127730051
## 37  -0.9614016205  0.677113381 -0.033065634 -0.308948236 -0.250023626
## 38  -0.6884528610  0.157409533 -0.038338941  0.101996502 -0.483773374
## 39  -0.8646348670  0.261522533  0.027088994  0.273715346 -0.666146176
## 40  -0.0320632153  0.976647523  0.650060348  0.447171577 -1.010621870
## 41   0.2948063760  0.328320739 -0.457093668 -0.054784499  0.023551074
## 42   0.1602610638  0.288702392 -0.472557922 -0.091667328 -0.094221523
## 43  -0.0637036321  0.322411175 -0.721714170 -0.140000142 -0.106846431
## 44  -0.5432267855  0.393365910 -1.150908630 -0.298545452 -0.369726695
## 45  -0.2771737297  1.053071651 -0.640151446  0.099728928 -0.401931398
## 46   0.9170711685 -0.749185536  0.067017671 -0.024822146 -0.069413250
## 47   0.2993916161 -0.510771764  0.117298536  0.419845552 -0.539632922
## 48   0.6229103192 -0.779776122 -1.283347660  0.525866477 -0.755684687
## 49   1.6367616601 -0.093132389 -0.459169515 -0.287272431 -0.255412483
## 50   0.8115322896  0.379357083  0.042966145 -0.797178449 -0.001084192
## 51   0.0694691478 -0.070920275 -0.184485962 -0.173005147 -0.198440668
## 52   0.3691362712 -0.109510292  0.021287314  0.261102004  0.040468167
## 53  -0.3434106096  0.325863703  0.606656320  0.076846401 -0.090805686
## 54   0.0694691478 -0.070920275 -0.184485962 -0.173005147 -0.198440668
## 55   1.2168826634 -0.263037412 -0.213770778 -0.087103051  0.207039728
## 56  -0.0296949830  0.625388100  0.133905329 -0.837124672  0.315605638
## 57   0.8840371721  0.077177558 -0.464930527 -0.667185212  0.020794706
## 58  -0.5771593965 -0.027046959  0.255651153 -0.070891410 -0.325220179
## 59   0.4929329280  0.361677028  0.249053650  0.069351972 -0.716028831
## 60   0.5317912348 -0.563884077  1.069788172 -0.917778633 -0.141296382
## 61   0.0549233337  0.236717720 -0.172217223 -0.081837619  0.172672742
## 62  -0.3115313288  0.516937761 -0.258984356 -0.221316897  0.127412404
## 63  -0.4083725776  0.233362056 -0.060951083 -0.293257887  0.001143481
## 64  -0.6967194754  0.182156410 -0.073026423 -0.240322981 -0.165579778
## 65  -0.3668212626 -0.852411009  1.400584325 -0.847040970  0.077156876
## 66  -0.1106580776  0.628458190 -0.186080728 -0.094760737  0.268914342
## 67  -0.3393240788  0.733470417 -0.141472370 -0.027545029  0.207561754
## 68  -0.6303889106  0.675721549  0.024826477 -0.149456305  0.361512075
## 69  -1.0240875385  1.132071113 -0.083765769 -0.121807119  0.289493307
## 70  -1.5273232787  0.570907171  0.351685856 -0.379070762 -0.350596156
## 71   0.7912856221 -0.893325072  1.180987550 -1.862895591 -0.122706854
## 72  -0.5842243780 -0.003322980  0.178131093  0.109562874  0.055520127
## 73  -0.0291019956  0.680692773 -0.337307495  0.007929935  0.014083476
## 74  -0.2354620812  0.850928858 -0.347814501 -0.023909405 -0.062507983
## 75  -0.5951094179  1.033249054 -0.396816441 -0.070007940 -0.078366877
## 76  -0.9271158460  1.455996081 -0.571408636  0.111684605 -0.240209168
## 77  -0.4411372935  0.342230508  0.299056070 -0.327418086  0.141102391
## 78  -0.2342929811  0.278218956 -0.349309742  0.208104928 -0.345496714
## 79  -0.0451935677 -0.586746114  0.386229474 -0.091788224 -1.080138436
## 80  -0.7497861942 -0.286591181  0.686257637  0.243239651  0.028561431
## 81  -1.1743001369 -1.105797449  1.472376135 -0.594520461  0.038954674
## 82  -0.7971611706  1.222157462 -0.108409117 -0.877229799  0.416944642
## 83   0.0685620493 -0.350584339  0.841022231  0.853972201  0.193036818
## 84  -0.9185248956 -0.104873940  0.395847783 -0.438836987  0.095269694
## 85  -0.0041278126  0.868545527 -0.722329136  0.866551034 -0.297719637
## 86   0.3940900111  0.927222090 -0.148348974 -0.030390397 -0.189020976
## 87   1.0099246222  0.374504205  0.077265951  0.261749831 -0.269144258
## 88  -0.0441403702 -0.227308122 -0.043071889 -0.224400290 -0.031334965
## 89   0.9244624594  0.363904444 -0.328409812  0.183615125 -0.478591509
## 90   0.2611447330  0.095723620 -0.113182966  0.178279594 -0.185322901
## 91   0.0899780220  0.273866464  0.657656201  0.026180969  0.202434750
## 92  -0.1378300388  0.158141295 -0.642096838  0.122899899  0.030015235
## 93   0.5869727328  0.894260366 -0.369274521  0.031691582 -0.361665958
## 94   0.8584757004  0.628220126 -0.286820377 -0.001420714 -0.323774219
## 95   0.7010071451  0.543611045 -0.865560169  0.014508715 -0.453482461
## 96  -0.1370061322  0.605976972 -0.191458241  0.038139195 -0.044344728
## 97   0.2019486311  0.230794490  0.072306613 -0.107952313 -0.048184545
## 98  -0.0239104754  0.257776059 -0.684364087  0.022736959 -0.123514613
## 99  -1.2393592819  0.707393276 -0.337450292 -0.591257888 -0.541464155
## 100 -1.0627931657  0.634613012 -0.468635810 -0.505603960 -0.384817887
## 101 -0.7295722955  0.262618641 -0.213696987 -0.651811809 -0.367348387
## 102 -1.0408094079  0.610265279 -0.522116158 -0.548131691 -0.551992607
## 103 -1.2264432666  0.600049267 -0.142727740 -0.654117100 -0.237737394
## 104  0.3201103919 -0.494089562  0.431089820  0.297046962  0.231877203
## 105 -0.0488928561 -0.276858971  0.362832861  0.347029302  0.125766216
## 106  0.0850177441 -0.257659160  0.333722606  0.289725788  0.084682384
## 107 -0.4391413734 -0.085113706 -0.255262746 -0.018571407  0.021580005
## 108  0.1295669538 -0.668444107 -0.564361146 -0.671138653 -0.109338870
## 109 -0.1404849728  0.004624673 -0.248875318 -0.394097702 -0.059433357
## 110  0.5822100547 -0.434402932 -0.623028673 -0.117980838 -0.161361845
## 111  0.4074993953 -0.444104090 -0.351193940 -0.059018453 -0.537737449
## 112 -1.0871656264 -0.410519649  0.288288822  0.658797567 -0.726477547
## 113 -1.3633495860 -0.543781040  0.548988447  0.840907166 -0.967781281
## 114  0.4491108556 -0.974013262 -0.244981446 -0.487906446  0.040430982
## 115  0.6025446941 -1.632018848 -0.080521316  0.881964127  0.594413526
## 116  0.0207082582 -1.997866528 -1.053242561  0.126606088  0.434543709
## 117  0.2173261634  0.243889157 -0.009302939  0.144911273  0.437998976
## 118  0.6742492971  1.044942265  0.450281766  0.457907363  0.473308263
## 119  0.1484518166  0.074596169  0.454103599 -0.561520420 -0.274113198
## 120 -0.0703921811 -0.156365438  0.946415301 -0.916743116 -0.588677184
## 121  1.4195735960  0.139852801  0.186073355  0.618395971 -0.529742424
## 122  1.2900980133  0.337235411  0.053750879  0.490527007 -0.369863820
## 123  1.0461977155  0.286884033  0.022228389  0.471534857 -0.397214451
## 124  0.0208799066  0.555979728 -0.273673971 -0.330394850 -0.159434269
## 125 -0.0598533483  0.445689085 -0.219633087 -0.385501057 -0.301748412
## 126 -0.2024070579  0.611845449 -0.300051938 -0.512764923 -0.314434876
## 127  0.1547990090 -0.981373535 -0.307309734 -0.361930282  0.043790535
## 128 -0.1413478622 -1.449021281 -0.114893771  0.125317843  1.378675822
## 129  1.0796418646 -1.228510957  0.096908018  0.220303341  0.725037834
## 130  0.9390326265 -1.613743005  1.161748443  1.091867979  1.198178189
## 131 -0.5049675572 -0.688444017  0.070928284 -0.172671517  0.744168650
## 132  0.3466809129  0.015470390 -0.159718046  0.637253387  0.711966302
## 133  0.0791218619 -0.268266802 -0.208318238  0.317382975  0.374364800
## 134  0.5068986650 -0.724087479  0.013623130  0.435165323  0.560597293
## 135  0.6479038947 -0.051704158 -0.247745683 -0.045343209  0.407749281
## 136 -0.0006929558 -1.178778664  0.255293282  0.251131969  0.823957870
## 137  0.2430843759 -0.225194680 -0.044147127  0.281102929  0.345339788
## 138  0.1939874214 -0.031551246  0.330552981 -0.139704245  0.020964268
## 139  1.3479717578 -0.784825667  0.343618194  0.032372652  0.288091955
## 140  0.4863037337  0.244597161  0.502910655  0.308588924  0.042799330
## 141  0.1470292394  0.632226062  0.952429745  0.549588841  0.389201719
## 142  0.3381631302 -0.520635492  0.285295605 -0.061717253 -0.028937494
## 143 -0.0066499797 -1.286401329  0.570554018  0.012952201  0.131068441
## 144 -0.0066499797 -1.286401329  0.570554018  0.012952201  0.131068441
## 145 -0.7869538902 -0.759128393  1.061799095  0.853336335 -0.198991347
## 146 -0.7047167143 -0.866139882  0.423871780  0.112120586 -0.256427570
## 147 -1.1310935543 -0.921305158  0.537183000  0.345595810 -0.228853088
## 148 -0.6286848374 -0.739592882  0.391263673  0.342149411 -0.602268038
## 149 -0.5805876722  0.882361833  1.442876741  1.321898453  0.518091291
## 150  0.3426502011  0.536381655  0.310849950 -0.099265985  0.024273196
## 151  0.3054332459  1.374305968  0.839048965  0.176603847  0.145679402
## 152 -0.9708857504  0.146475213  0.640469703  0.298997217  0.036164264
## 153 -0.0854585191 -0.782520131  0.655843306 -0.092190829  0.356351438
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## 365 -0.1171637338  0.392750377 -0.603393631  0.215015026 -0.209873599
## 366  0.5122469693  0.546711676 -0.689744817  0.228795983 -0.651182530
## 367 -0.2400749304  0.587774827 -0.125269798 -0.415386851 -0.093693000
## 368  0.0674857376 -0.274035052  1.265883661 -1.310906552 -0.866200330
## 369  1.1710913746 -0.871552193  0.512799437 -0.182118487  0.294065039
## 370  1.3749629598 -0.015038789 -0.008223591  0.001604537  0.057154405
## 371  0.4923397528 -0.125196064 -0.628245951  0.521457762  0.742531944
## 372  0.0971560409  1.073428897  0.387105505  0.005057744  0.269708674
## 373  1.3059858460  0.378706961 -0.015155300  0.328363489  0.095765143
## 374  0.9946486381 -0.433954592 -0.177639578  0.192759917  0.150604725
## 375  0.0372928443  0.238941993  0.546299821  0.165658360  0.379001059
## 376  0.6220563233  0.395517856  0.421809799  0.002142859 -0.306699785
## 377  0.0520700948  0.038761814 -0.589250106  0.195338130  0.302276624
## 378  0.9184556058 -0.048970841  0.170753719  0.181267310  0.301488273
## 379  0.0864128447  1.055404123  0.055754823 -0.994280376 -0.334121750
## 380  1.2115115406 -0.233724669  0.275348313  0.305698304  0.517989931
## 381  1.4973610098 -0.370933510  0.261124339  0.234558728  0.510441684
## 382  2.2245902437  0.465174975  0.161991398  0.055826383  0.160233946
## 383  0.7533310719  0.565753326 -0.019165423 -0.493019766 -0.331825680
## 384  1.0217534902  0.653109737 -0.209086149 -0.011917868 -0.088042610
## 385  1.6290102361  0.254682481  0.498865174  0.791897764  0.532058873
## 386  0.4558973907  0.118267128  0.262849223 -0.024019440  0.188369580
## 387  0.8857583943  0.592446074 -0.043988692  0.182374132  0.038895835
## 388  0.5299949621 -0.006094956  0.285242187  0.131425808  0.354914968
## 389  1.7221727762 -0.119045849 -0.035705302 -0.251304828  0.200657670
## 390  1.2956436056 -0.308644833 -0.740801138 -0.145285428 -0.280090081
## 391  1.0774727127  0.466869315 -0.036048593  0.329988277  0.070056438
## 392  1.0975918723  0.464332009 -0.011523644  0.349593878  0.047197759
## 393  1.5316715827  0.380340157 -0.421800531  0.260209444 -0.205912622
## 394 -0.4866982705  0.590118107  0.143052323  1.618513471 -0.312425442
## 395 -2.4276398955  0.296094801  0.568260430 -0.511701206 -0.161538421
## 396 -0.9822604976 -0.186389852  0.295494023  1.969560149  0.031541383
## 397 -0.4825387196 -0.906631665  0.691578061 -0.346112809 -1.330248399
## 398 -0.9333167773 -1.070294056  1.142793236 -1.167400569 -1.204673191
## 399 -0.9243676371 -1.497046368  1.252123775 -1.367264960 -0.954937344
## 400 -0.9714637990 -1.525961209  1.221956458 -1.380957308 -0.885181758
## 401  0.6313733374  0.728007312  0.365704478  0.315483202 -0.105412501
## 402  0.5817692224  1.471865150  0.461636937 -0.009296875  0.344018935
## 403  0.5850659423  0.448976893 -0.017100287 -0.313712236  0.173734737
## 404  0.4290880909  0.664846799  0.149697047 -0.219986417  0.206832663
## 405  0.3783104072  0.110603065  0.098309074 -0.156699424  0.122732681
## 406 -0.0714739805 -0.383239288  0.246122293  0.368679963 -0.473539923
## 407  0.2989792611  0.457039617  0.344885399 -0.038295920 -0.035311217
## 408  0.3104505271  0.353742581  0.054219307 -0.273066661  0.487652873
## 409  0.0543784420  0.414520607  0.215294661 -0.311028615  0.206520681
## 410  0.1545687327  0.289794792  0.230568805 -0.163967752  0.226682063
## 411  0.3199801127  0.258578652  0.605654947 -0.247863162 -0.250732627
## 412  0.3913342918  0.197460426  0.563008803 -0.248323214 -0.342879590
## 413  0.9803314015 -0.231930122  0.542346074  0.114249726 -0.442365573
## 414  0.3589499027 -0.198215537  0.056472909  0.191202841 -0.303302368
## 415  0.2693647156  0.851903902  0.380014977  0.332453720 -0.087242004
## 416 -1.0252626259 -0.326416388  0.454984675 -0.181768888  0.032363594
## 417 -1.2975217722 -0.403062508  0.421060628 -0.097363085  0.020084956
## 418 -0.3225610613  0.020348084  0.699157983 -0.324540322  0.172313984
## 419  0.5246148918  0.259537599  0.085100019 -0.608815369  0.360294509
## 420 -0.2919062786 -0.580780694  0.171156178  0.267173130 -0.302886805
## 421 -0.1531122584 -0.361864852  0.347076353  0.458444501 -0.542438002
## 422  0.3639158518 -0.109346621  0.066354716  0.227025797  0.036155703
## 423 -0.3417785242 -0.394748791  0.160665292  0.306032595 -0.456762080
## 424  0.5523774122  0.968355284  0.735284609  0.444516948  0.762073438
## 425 -0.3660582730  0.338436437  0.761241843  0.159673961  0.592441773
## 426 -0.2172835497  0.585120248  0.813929791  0.218099261  0.259362560
## 427 -1.0549220009 -0.313687264  0.887805393 -0.010540163  0.278442340
## 428 -1.0999616013 -0.238844708  0.677921079  0.156265740  0.412697593
## 429 -0.4496562640  0.169991754  0.924713109 -0.051989630  0.548862571
## 430 -0.2440585742 -0.026411556  0.230341867 -0.115664417  0.140897050
## 431  0.9400143370  0.573692918  0.404362084  0.154959455 -0.025110770
## 432  0.1186813909  0.246613828  0.206975256 -0.439103384  0.042111675
## 433  0.8829273557  0.768117512  0.637814869  0.074047200 -0.012735111
## 434 -0.8268702083 -0.084927087  0.116872898  0.204975210 -0.317519079
## 435 -0.0868684980  0.308784585 -0.113934626 -0.097932987  0.203420292
## 436 -0.5554445732 -0.134289251 -0.265820208  0.148314869 -0.044852063
## 437 -0.3281900727 -0.116639671 -0.411345196 -0.085557309  0.205933054
## 438 -0.4421919888  0.042538668 -0.402502945 -0.019155110  0.112796283
## 439 -0.3876530271 -0.059878562 -0.464243180 -0.093765014  0.160731672
## 440  0.0491444745  0.491684594 -0.300395816 -0.322088404  0.250746661
## 441  0.1036834362  0.389267365 -0.362136052 -0.396698307  0.298682050
## 442  0.4244014687 -0.346025684 -0.157383679 -0.675655745  0.560036613
## 443  0.1841866866  0.025890469  0.157956223 -0.668552886  0.343788968
## 444 -0.1120587248  0.104741279 -0.320480483 -0.566087184  0.290510718
## 445 -0.0168412832  0.519689419 -0.160425116 -0.239482869  0.341117734
## 446 -0.4642721817  0.035916241 -0.330771483  0.046334964  0.129442492
## 447 -0.3795238981 -0.087742274 -0.383078651 -0.003979506  0.187971697
## 448 -0.5197163535 -0.321254187 -0.218246794  0.209993654 -0.119019432
## 449 -0.0145538465  0.520797575 -0.169927901 -0.247762124  0.334651323
## 450 -0.5610838981 -0.103315515 -0.234140558  0.130641699 -0.025957106
## 451 -0.4117364269 -0.030862533 -0.376457837 -0.060725989  0.195986634
## 452 -0.3485700299 -0.127052235 -0.178106481  0.018218800  0.142374043
## 453 -0.5106197438  0.068577195 -0.089771799  0.108178604  0.037606808
## 454 -0.2420510054 -0.257465584 -0.176480036  0.311606059 -0.286809785
## 455 -0.3118390492 -0.310500618 -0.413686875 -0.133591687 -0.270360257
## 456 -0.2179582214 -0.418765269 -0.379030030  0.009523310 -0.248566291
## 457 -0.0397627470 -0.373626587 -0.540490105 -0.069116468 -0.097099810
## 458  0.8547641068 -0.135387142 -0.116887976 -0.742627055  0.058419124
## 459  0.2637764597  0.086430561 -0.438156073 -0.284129906  0.620092030
## 460  0.0582921321 -0.048348815 -0.432499692 -0.161165198  0.495392845
## 461  0.9440795874  0.066962148 -0.079410377 -0.473719889  0.262623307
## 462  0.4749307134  0.675633137  0.230630479 -0.169034634  0.155269317
## 463  0.0255256948  0.531281948  0.352460220 -0.193738862  0.095563861
## 464  0.1193792320  0.424651773  0.388623081 -0.051089384  0.117015565
## 465 -0.6360876691  0.046261898  0.102370058  0.099170431 -0.229393074
## 466 -0.4071369572  0.059150889 -0.317268565  0.040368626  0.010175806
## 467 -0.5352483030 -0.051580853  0.175323291  0.239478317 -0.157447945
## 468 -0.3846982860  0.010253155 -0.016672403  0.132877798  0.045354162
## 469  0.5868958416  0.324325841 -0.313977541 -0.011171527  0.037441728
## 470  0.4806664341  0.418388081 -0.177643972  0.074854823  0.024122521
## 471  0.3996488434  0.489946869 -0.166430133  0.102888042 -0.022541563
## 472  0.4829538708  0.419496236 -0.187146757  0.066575568  0.017656110
## 473  0.0266344339  0.542980201  0.064735874 -0.831659724  0.704795689
## 474  0.0317525046 -0.255491358 -0.276364815 -0.187777758  0.092282197
## 475 -0.3471756631 -0.347586326 -0.058427872 -0.139557400 -0.232044698
## 476 -0.2533221259 -0.454216501 -0.022265011  0.003092078 -0.210592994
## 477  1.6145949147 -0.220703213 -0.402270888 -0.446314646  0.243208204
## 478  0.9098718413 -0.312352250 -0.790397662 -0.320694071 -0.512187247
## 479  0.2830689304  0.051669576 -0.622456951 -0.263150782  0.049817659
## 480  0.2391457426  0.037627389 -0.284589469  0.054391310 -0.253183289
## 481  0.1184242266  0.063723212 -0.241272512 -0.391659672  0.396228814
## 482 -0.1380225232 -0.335618222  0.069850940  0.272071882 -0.220443156
## 483 -0.4704415839 -0.059467751 -0.189981811  0.193595891  0.107776339
## 484  0.0624577814 -0.224876782 -0.074309922  0.151009508 -0.292528138
## 485 -0.0435707811 -0.766914018 -0.126644217  0.412791229 -0.393260137
## 486 -0.0017391123  0.614250735  0.170250639 -0.352415625  0.156022971
## 487  0.1123248693  0.643459190  0.083147348 -0.412684332  0.070055366
## 488  0.6694365119  0.258027999  0.146268821 -0.019746762 -0.160417100
## 489  0.2352503840  0.367285238  0.082065426  0.386339117 -0.066425153
## 490 -0.5482018655 -0.138173280 -0.063134876  0.231090034 -0.051155275
## 491 -0.0974409831  0.574615477  0.219038376  0.236188277  0.021771816
## 492  0.3091141016 -0.749229071 -0.408765377  0.108031291  0.043213986
## 493  0.2693647156  0.851903902  0.380014977  0.332453720 -0.087242004
## 494 -0.5853304736 -0.650080216 -0.203575915  0.528105054 -0.441317558
## 495 -1.2961511328 -0.159951301 -0.115519205 -0.107510064  0.583049167
## 496  0.5616544429 -0.238445319  0.045565441 -0.143077517 -0.196921840
## 497  0.1473609993  0.717526718 -0.275119046 -0.359653429  0.497584487
## 498  0.8831254546  0.723917290  0.037192893 -0.118188558  0.197604075
## 499  0.1067956159 -0.666281710  0.131931214  0.384871143 -0.283450519
## 500  0.5792944614  0.808270903 -0.344283350  0.005921737 -0.244162483
## 501 -0.0281559952  0.684933247  0.268564511  0.413636985  0.038398709
## 502 -1.0715288379 -0.077500069  0.077457941 -0.367115791  0.150207210
## 503 -1.0571735305 -0.031327049  0.043678932 -0.273964305  0.102886380
## 504 -0.5430040752  0.246610342  0.140051607 -0.148384250  0.219136440
## 505 -0.7152373039 -0.001971726  0.021097652  0.200239410 -0.120763164
## 506 -0.2753969395 -0.488274518  0.001104044  0.324771870 -0.473954162
## 507 -0.2305187554  0.082019108 -0.228406463  0.098436641 -0.031237172
## 508 -0.7111911548  0.273731096  0.252574312 -0.061970346  0.427627123
## 509 -0.0098969088  0.484446467  0.274583938  0.374891889  0.044856105
## 510  0.7339666182  0.179222830  0.493633341 -0.584271028  0.135396501
## 511  0.9465900171 -0.300564597  0.586239485 -0.367019968  0.179848539
## 512  0.8831254546  0.723917290  0.037192893 -0.118188558  0.197604075
## 513  1.2157187917  0.509549779 -0.415544317 -0.226230333 -0.117094431
## 514  0.3179171839 -0.140974732 -0.521964694 -0.671699938 -0.198300515
## 515  1.0020843135  0.449639332 -0.083291045 -0.132296766 -0.312458002
##           Dim.11        Dim.12        Dim.13        Dim.14
## 1   -0.065237310  0.2504691510  0.0818017284  1.426266e-03
## 2   -0.384315637 -0.3506123484 -0.0320794779  2.613573e-02
## 3    0.458934810 -0.3012400810  0.0487860533  5.188028e-04
## 4   -0.235871692  0.0834160069  0.0415733564  1.291964e-02
## 5   -0.167649419 -0.1346505723 -0.0202867814  3.437834e-02
## 6   -0.057339431  0.0956369628  0.0208889510 -1.304307e-03
## 7   -0.041917423  0.0467175968 -0.0248796559 -3.389359e-02
## 8   -0.162269897  0.0400447953  0.0221645422 -2.139312e-02
## 9   -0.237755047  0.1454230508  0.0210805994 -1.607561e-04
## 10  -0.282849553  0.0603844034  0.1400948479 -2.608737e-02
## 11  -0.029966991  0.0660758790 -0.0111843453  4.723220e-02
## 12   0.072193521  0.0028701823  0.0719913217 -1.297762e-02
## 13  -0.177645272  0.3230602853  0.1161064783  1.062694e-02
## 14  -0.103090833  0.0329252102  0.0250110061 -1.973892e-02
## 15  -0.284500729 -0.0736533087 -0.0649717697 -9.649971e-03
## 16  -0.190933687  0.3220399060  0.0589676946  3.079857e-02
## 17  -0.188388773 -0.2126769961 -0.0129660288  2.917370e-02
## 18  -0.273634233  0.1427162386  0.0153449619  4.178824e-03
## 19   0.311727579  0.0491058914  0.0423162168 -4.397794e-03
## 20  -0.323146734  0.0401307472  0.0021609745 -2.031389e-02
## 21  -0.083847162  0.0698758116  0.0566092018 -2.626110e-03
## 22  -0.382095926  0.0026376418 -0.0404291173  1.247144e-02
## 23  -0.184255679  0.0253501000  0.0614756418  2.356499e-02
## 24  -0.100831664  0.3051254690  0.0561990779 -9.689815e-04
## 25   0.119001660  0.1455678210  0.0409403768 -2.095430e-02
## 26  -0.092498383 -0.0229644347  0.0051070762 -2.077034e-02
## 27   0.161941405 -0.1319213262 -0.0155466990  3.004091e-02
## 28  -0.127925562  0.0637960984  0.0383862244 -1.225987e-02
## 29   0.227939866  0.1290609272  0.0884946019  4.922036e-02
## 30   0.182390252  0.4105327875  0.1223161776  2.200188e-02
## 31   0.095845650  0.2030317688  0.0437556955  3.419229e-02
## 32  -0.190602681 -0.2193689002  0.0042870329  4.173212e-02
## 33  -0.190427809  0.1971689958  0.0369954884  2.130559e-02
## 34   0.141501649  0.0136522979  0.0049105794  2.250189e-03
## 35   0.086032940 -0.1328634643 -0.0020200045  1.363532e-03
## 36   0.205972601 -0.1653855737  0.0536673094 -1.073060e-02
## 37   0.418656704 -0.2352949708  0.0757733768 -3.139832e-02
## 38   0.702513162  0.0739809692 -0.2658391426 -3.438428e-02
## 39   0.037532803  0.0081747130 -0.2014605191 -4.746996e-02
## 40  -0.956964463 -0.0387827712  0.2666044807 -3.251066e-02
## 41  -0.121875038 -0.0433643514 -0.0089180406  5.910839e-03
## 42  -0.095774501  0.0048207025 -0.0019552023 -1.885280e-02
## 43  -0.102182713  0.0629806343  0.0571204004 -1.228302e-02
## 44  -0.126085902  0.0811035136  0.1280176016  1.539747e-02
## 45  -1.144072870 -0.0708606352  0.2105409295 -3.998524e-02
## 46   0.118607302  0.1430516259  0.0643205520  1.129435e-02
## 47  -0.113831320  0.4249961144  0.1743085722  2.571877e-02
## 48  -0.078251424  0.1638376351  0.2504302624  3.865419e-02
## 49   0.207803118 -0.2399828317 -0.0342778908  5.401244e-02
## 50   0.323196870  0.1032109342  0.0756123510  1.804208e-02
## 51  -0.303556699 -0.0197208371  0.0470698796 -1.631779e-02
## 52  -0.218966455 -0.0602792088 -0.0047195766  4.719092e-03
## 53   0.142994300 -1.0924765787 -0.1654079305 -5.475662e-02
## 54  -0.303556699 -0.0197208371  0.0470698796 -1.631779e-02
## 55   0.037257530 -0.0472498991 -0.0427105496  1.243824e-02
## 56  -0.003145038  0.2912730359  0.1097335214 -2.837567e-02
## 57  -0.012392315  0.0804485070  0.0450374944  3.893945e-02
## 58  -0.529091861 -0.1807713761  0.0052376196 -1.385753e-02
## 59  -0.343473552 -0.0853631938  0.0239291104  3.273968e-03
## 60   0.075718933  0.1770732424  0.0105396408  3.804249e-03
## 61  -0.238299400 -0.2326024257 -0.0778862616 -9.962302e-03
## 62  -0.066491281 -0.1345878485  0.0372499262 -4.420194e-04
## 63   0.134811871 -0.0443134572  0.0240235650  3.524717e-03
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## 451 -0.166990547  0.1352747509 -0.0286089641  1.891438e-03
## 452 -0.193990093  0.1574138331 -0.0218946327  2.566645e-03
## 453 -0.103085381  0.0773686068 -0.0286273256 -8.551801e-03
## 454  0.018852454  0.1165786138  0.0183902145  2.173999e-02
## 455  0.082361904  0.0758113496  0.0254082530  2.543763e-04
## 456  0.131522811  0.0022356150 -0.0104504552  4.807387e-02
## 457  0.078910143 -0.0215115491 -0.0290486428  1.131208e-02
## 458  0.270275767 -0.0820440832  0.0003951911 -1.358322e-02
## 459  0.123424738  0.0314434463 -0.0122489122 -9.675734e-03
## 460 -0.012589153  0.0106489961 -0.0164938244 -5.654435e-03
## 461  0.285181534 -0.0577284210  0.0200823242 -1.275697e-02
## 462 -0.029243162  0.0161530711 -0.0283643006 -5.981094e-05
## 463 -0.009066621  0.1167267647  0.0063030333  4.513201e-02
## 464  0.040825753  0.0444030455 -0.0246154243 -7.158270e-04
## 465  0.143631433  0.2265184197  0.0394470149  5.151789e-03
## 466  0.142649514  0.2083301142  0.0442319821  1.118139e-03
## 467  0.030576228  0.1011278000 -0.0182215744 -2.647563e-03
## 468 -0.019650685  0.1410149341  0.0024945827 -1.176240e-02
## 469  0.035651606 -0.0657366023  0.0333301400  4.632990e-03
## 470 -0.017688841 -0.0270056933  0.0212105539 -4.172639e-03
## 471 -0.049889713 -0.0129587787  0.0282936587 -5.526235e-03
## 472 -0.020761043 -0.0281396751  0.0215002424 -4.117958e-03
## 473  0.262186824 -0.0300496371 -0.0152962043  1.200925e-02
## 474 -0.073770238 -0.0198733622 -0.0428823099 -2.648880e-02
## 475 -0.129998647  0.0868988593 -0.0033020578  4.443303e-03
## 476 -0.080106273  0.0145751401 -0.0342205155 -4.140454e-02
## 477  0.223430951  0.1017132138  0.0118668453 -2.767874e-02
## 478  0.109834322  0.0746548744  0.0348208693  6.513495e-02
## 479  0.050952463  0.1234732574  0.0242337722  2.631956e-02
## 480  0.301572306  0.0936417963  0.0469647324  4.466470e-02
## 481 -0.009560463 -0.0424101212 -0.0504340493 -2.276838e-02
## 482 -0.247368833 -0.0114891492 -0.0288579565 -9.596090e-03
## 483  0.025351928  0.0880750249 -0.0046975146  1.068105e-02
## 484 -0.080912558  0.0963155128 -0.0138667965 -7.857365e-04
## 485  0.024447248 -0.0058079628 -0.0210368148  1.996508e-02
## 486  0.204608899  0.3458443699  0.0763448650  2.396325e-02
## 487  0.254428557  0.1538084330  0.0516446817  2.844271e-02
## 488  0.167541395  0.1707116760  0.0444479326  2.285206e-02
## 489 -0.243038680 -0.0762936548 -0.0117673015 -2.126885e-02
## 490 -0.270543713 -0.0349030150 -0.0539615032 -1.499666e-02
## 491 -0.323427808 -0.0405215752  0.0238371120 -1.989182e-02
## 492 -0.067341388  0.0268335735 -0.0243796906  3.533543e-02
## 493 -0.129065116  0.0523164738  0.0199237522 -1.467302e-02
## 494  0.187518172  0.1855566419  0.0355304643  4.918491e-02
## 495  0.034751701  0.0332740584 -0.0316897002  2.400620e-02
## 496 -0.296397308  0.1309233639 -0.0131403904  9.936291e-03
## 497  0.097482190 -0.0869061381 -0.0353948865  7.646734e-03
## 498  0.148183821  0.1443510394  0.0813640096  1.263986e-02
## 499 -0.203798685  0.0774293489 -0.0156982729  1.094306e-02
## 500  0.246920784 -0.0583854444  0.0644087517 -2.938786e-02
## 501  0.074329557  0.1070778489  0.0214551051  2.322682e-02
## 502 -0.088315475  0.0074712823 -0.0051199169  7.754231e-03
## 503 -0.152532565  0.0145455076 -0.0036840400 -4.243227e-02
## 504  0.096440342 -0.0231539751  0.0155132171  2.974930e-02
## 505  0.112361459  0.0740098839 -0.0025275301 -7.609642e-03
## 506  0.055743313  0.0337660334  0.0351842009  8.064868e-03
## 507  0.127400234  0.0599337150  0.0099825809  7.713804e-03
## 508  0.058420558 -0.0249291824 -0.0253657832  3.571779e-02
## 509 -0.287997511 -0.1088842771 -0.0255634175 -2.285171e-02
## 510 -0.053433972  0.0927224765  0.0152862533  7.144200e-03
## 511  0.095148721  0.1284755661  0.0175837962 -3.256034e-04
## 512  0.148183821  0.1443510394  0.0813640096  1.263986e-02
## 513  0.260763763  0.0326680198  0.0658802214  3.204732e-02
## 514  0.104948383 -0.0169458497  0.0526636917  4.044229e-02
## 515  0.413400573 -0.0034841826  0.0514136221  4.769635e-03

Visualisasi Data

library(factoextra)

## Warning: package 'factoextra' was built under R version 4.4.3

## Loading required package: ggplot2

## 
## Attaching package: 'ggplot2'

## The following objects are masked from 'package:psych':
## 
##     %+%, alpha

## Welcome! Want to learn more? See two factoextra-related books at https://goo.gl/ve3WBa

fviz_eig(pca_result, 
         addlabels = TRUE, 
         ncp = p, 
         barfill = "skyblue", 
         barcolor = "darkblue", 
         linecolor = "red")

fviz_pca_biplot(pca_result, 
                geom.ind = "point", 
                addEllipses = TRUE, 
                )

contrib_circle <- fviz_pca_var(pca_result, col.var = "contrib",
                               gradient.cols = c("#00AFBB", "#E7B800", "#FC4E07"), 
                               repel = TRUE) + 
  ggtitle("Kontribusi Variabel")
plot(contrib_circle)

fviz_contrib(pca_result, choice = "var", axes = 1) + ggtitle("PC 1")

seperti ini

fviz_contrib(pca_result, choice = "var", axes = 2) + ggtitle("PC 2")

seperti ini

fviz_contrib(pca_result, choice = "var", axes = 3) + ggtitle("PC 3")

seperti ini Factor Analysis

varcov = cov(scale_data)
pc = eigen(varcov)
pc$values

##  [1] 6.816508193 2.604247876 1.435911259 0.820125615 0.664797468 0.467602025
##  [7] 0.399717867 0.260826187 0.215474735 0.145291385 0.094497476 0.043493563
## [13] 0.026143640 0.005362711

pc$vectors

##              [,1]         [,2]        [,3]        [,4]         [,5]        [,6]
##  [1,] -0.36949057  0.001368424 -0.11779416  0.04143723  0.116329233 -0.02441011
##  [2,] -0.34540359 -0.165402558  0.05993035  0.13014083  0.224890630 -0.09088434
##  [3,] -0.34537867 -0.162511046  0.05731728  0.12974522  0.218931238 -0.10031320
##  [4,] -0.32865253 -0.156014542  0.23810568  0.22973370 -0.069592999 -0.04006214
##  [5,] -0.24054731 -0.231763871  0.29932287  0.25958106 -0.330015924 -0.35916228
##  [6,] -0.32677213 -0.148937976  0.14838810 -0.34640634 -0.002603662  0.21534372
##  [7,] -0.31543430  0.074924069 -0.22665125 -0.18386601  0.243385248  0.13423113
##  [8,] -0.25373980  0.275119483 -0.42614275  0.16379995 -0.036899773 -0.13612374
##  [9,] -0.09796561  0.469618235 -0.24899949  0.25854031  0.282715130 -0.26025096
## [10,] -0.21938954  0.254035276 -0.08794738 -0.34124433 -0.636223548 -0.38271456
## [11,] -0.32778691  0.011107884 -0.01992010 -0.38876491  0.006078372  0.34414225
## [12,] -0.01439324  0.341003319  0.60405821  0.04474178  0.273260507 -0.12267613
## [13,] -0.01990642  0.477681518  0.35792357 -0.33550559  0.107212497 -0.08370616
## [14,] -0.15023891  0.367460061  0.12183502  0.46338251 -0.377745102  0.64254751
##              [,7]        [,8]         [,9]       [,10]       [,11]       [,12]
##  [1,]  0.06366507 -0.08788376  0.046301085 -0.32884533 -0.05047819  0.11968035
##  [2,]  0.25953516 -0.29959462 -0.063269911 -0.01850633 -0.19693010 -0.07476747
##  [3,]  0.26171356 -0.30816262 -0.062734724 -0.02595621 -0.22993039 -0.06297561
##  [4,]  0.02376597  0.04546763 -0.007883432  0.37177870  0.69386350 -0.33625847
##  [5,] -0.53374224  0.24414120 -0.199441569 -0.17073871 -0.25875023  0.09704868
##  [6,] -0.17935472 -0.13063633  0.283935773  0.16078493  0.17896358  0.68936514
##  [7,]  0.06502865  0.55764149 -0.485842416  0.37946129 -0.18028262  0.06654421
##  [8,]  0.07568636  0.20953169 -0.002761977 -0.50588738  0.41603502  0.15284151
##  [9,] -0.44439392 -0.18572532  0.301922839  0.39525682 -0.11704566 -0.01420631
## [10,]  0.33803264 -0.05297244  0.132424587  0.22043391 -0.15211194 -0.06635545
## [11,] -0.32361011  0.08356730  0.298033096 -0.23649455 -0.12073453 -0.58052754
## [12,]  0.28337394  0.44056431  0.362712609 -0.10728263 -0.08203656  0.04968885
## [13,] -0.17496221 -0.35531213 -0.545312414 -0.15278152  0.18445580 -0.02561239
## [14,]  0.07166109 -0.09655347 -0.083456127  0.04591554 -0.16619525  0.07666342
##               [,13]        [,14]
##  [1,]  0.8317886085 -0.054317923
##  [2,] -0.1942224052  0.728298931
##  [3,] -0.2938436638 -0.681869084
##  [4,]  0.1240148051 -0.030286425
##  [5,] -0.0600152593  0.012314098
##  [6,] -0.1399889609  0.009323147
##  [7,]  0.0169994710 -0.003208779
##  [8,] -0.3593959354  0.020816478
##  [9,]  0.0093325414  0.002324827
## [10,]  0.0334594911  0.006394181
## [11,] -0.1198152795  0.004626427
## [12,] -0.0152567856  0.001111003
## [13,]  0.0009010252 -0.001589051
## [14,] -0.0152486082  0.003725770

sp = sum(pc$values[1:3])

L1 = sqrt(pc$values[1])*pc$vectors[,1]
L2 = sqrt(pc$values[2])*pc$vectors[,2]
L3 = sqrt(pc$values[3])*pc$vectors[,3]

L = cbind(L1,L2,L3)
L

##                L1          L2          L3
##  [1,] -0.96468237  0.00220832 -0.14115217
##  [2,] -0.90179500 -0.26692139  0.07181424
##  [3,] -0.90172995 -0.26225516  0.06868301
##  [4,] -0.85806060 -0.25177131  0.28532088
##  [5,] -0.62803159 -0.37401317  0.35867714
##  [6,] -0.85315116 -0.24035137  0.17781274
##  [7,] -0.82354985  0.12091008 -0.27159509
##  [8,] -0.66247512  0.44397908 -0.51064479
##  [9,] -0.25577296  0.75785498 -0.29837488
## [10,] -0.57279195  0.40995406 -0.10538692
## [11,] -0.85580060  0.01792555 -0.02387016
## [12,] -0.03757851  0.55030032  0.72384002
## [13,] -0.05197256  0.77086725  0.42889808
## [14,] -0.39225042  0.59299537  0.14599431

library(psych)
fa <- fa(r = scale_data, 
         covar = TRUE,
         nfactors = 3, 
         rotate = "varimax")

## Warning in fa.stats(r = r, f = f, phi = phi, n.obs = n.obs, np.obs = np.obs, :
## The estimated weights for the factor scores are probably incorrect.  Try a
## different factor score estimation method.

load <- fa$loadings

plot(load[,c(1,3)],type="n") 
text(load[,c(1,3)],labels=names(data),cex=.7)

fa.diagram(load)

Note that the echo = FALSE parameter was added to the code chunk to prevent printing of the R code that generated the plot.