Tugas Pengantar Analisis Multivariat 24B
Ayu Wulan Anggraeni Putri
2026-02-12
1. Deskripsi Dataset
Dataset Concrete terdiri dari 1030 observasi dan 9 variabel numerik yang berkaitan dengan komposisi campuran beton dan kuat tekan beton (MPa). Variabel meliputi kandungan semen (cement), blast furnace slag, fly ash, air (water), superplasticizer, agregat kasar, agregat halus, umur beton (age), dan kuat tekan beton (strength). Data ini digunakan untuk melihat hubungan antar komponen campuran beton serta pengaruhnya terhadap kuat tekan beton.
2. Import Data
# Install package jika belum ada
if(!require(readxl)) install.packages("readxl")## Loading required package: readxlif(!require(corrplot)) install.packages("corrplot")## Loading required package: corrplot## corrplot 0.95 loadedlibrary(readxl)
library(corrplot)
# Import data
data_concrete <- read_excel("Concrete_Data.xls")
# Ambil hanya variabel numerik
data_numeric <- data_concrete[, sapply(data_concrete, is.numeric)]
# Struktur data
str(data_numeric)## tibble [1,030 × 9] (S3: tbl_df/tbl/data.frame)
## $ Cement (component 1)(kg in a m^3 mixture) : num [1:1030] 540 540 332 332 199 ...
## $ Blast Furnace Slag (component 2)(kg in a m^3 mixture): num [1:1030] 0 0 142 142 132 ...
## $ Fly Ash (component 3)(kg in a m^3 mixture) : num [1:1030] 0 0 0 0 0 0 0 0 0 0 ...
## $ Water (component 4)(kg in a m^3 mixture) : num [1:1030] 162 162 228 228 192 228 228 228 228 228 ...
## $ Superplasticizer (component 5)(kg in a m^3 mixture) : num [1:1030] 2.5 2.5 0 0 0 0 0 0 0 0 ...
## $ Coarse Aggregate (component 6)(kg in a m^3 mixture) : num [1:1030] 1040 1055 932 932 978 ...
## $ Fine Aggregate (component 7)(kg in a m^3 mixture) : num [1:1030] 676 676 594 594 826 ...
## $ Age (day) : num [1:1030] 28 28 270 365 360 90 365 28 28 28 ...
## $ Concrete compressive strength(MPa, megapascals) : num [1:1030] 80 61.9 40.3 41.1 44.3 ...Pada bagian ini, terlebih dahulu dipastikan bahwa package
readxl dan corrplot sudah terpasang agar data
Excel dapat dibaca dan divisualisasikan. Data kemudian diimpor
menggunakan fungsi read_excel(). Karena analisis korelasi
dan kovarians hanya dapat dilakukan pada data numerik, maka dipilih
hanya variabel yang bertipe numerik. Struktur data ditampilkan untuk
memastikan bahwa semua variabel sudah terbaca dengan benar.
3. Statistik Deskriptif
summary(data_numeric)## Cement (component 1)(kg in a m^3 mixture)
## Min. :102.0
## 1st Qu.:192.4
## Median :272.9
## Mean :281.2
## 3rd Qu.:350.0
## Max. :540.0
## Blast Furnace Slag (component 2)(kg in a m^3 mixture)
## Min. : 0.0
## 1st Qu.: 0.0
## Median : 22.0
## Mean : 73.9
## 3rd Qu.:142.9
## Max. :359.4
## Fly Ash (component 3)(kg in a m^3 mixture)
## Min. : 0.00
## 1st Qu.: 0.00
## Median : 0.00
## Mean : 54.19
## 3rd Qu.:118.27
## Max. :200.10
## Water (component 4)(kg in a m^3 mixture)
## Min. :121.8
## 1st Qu.:164.9
## Median :185.0
## Mean :181.6
## 3rd Qu.:192.0
## Max. :247.0
## Superplasticizer (component 5)(kg in a m^3 mixture)
## Min. : 0.000
## 1st Qu.: 0.000
## Median : 6.350
## Mean : 6.203
## 3rd Qu.:10.160
## Max. :32.200
## Coarse Aggregate (component 6)(kg in a m^3 mixture)
## Min. : 801.0
## 1st Qu.: 932.0
## Median : 968.0
## Mean : 972.9
## 3rd Qu.:1029.4
## Max. :1145.0
## Fine Aggregate (component 7)(kg in a m^3 mixture) Age (day)
## Min. :594.0 Min. : 1.00
## 1st Qu.:731.0 1st Qu.: 7.00
## Median :779.5 Median : 28.00
## Mean :773.6 Mean : 45.66
## 3rd Qu.:824.0 3rd Qu.: 56.00
## Max. :992.6 Max. :365.00
## Concrete compressive strength(MPa, megapascals)
## Min. : 2.332
## 1st Qu.:23.707
## Median :34.443
## Mean :35.818
## 3rd Qu.:46.136
## Max. :82.599Statistik deskriptif dihitung menggunakan fungsi
summary() untuk melihat gambaran umum dari setiap variabel.
Output ini menunjukkan nilai minimum, maksimum, mean, median, serta
kuartil. Dari hasil tersebut dapat terlihat bahwa setiap variabel
memiliki rentang nilai yang berbeda, sehingga menunjukkan adanya
perbedaan skala antar variabel.
4. Correlation Matrix
cor_matrix <- cor(data_numeric)
cor_matrix## Cement (component 1)(kg in a m^3 mixture)
## Cement (component 1)(kg in a m^3 mixture) 1.00000000
## Blast Furnace Slag (component 2)(kg in a m^3 mixture) -0.27519344
## Fly Ash (component 3)(kg in a m^3 mixture) -0.39747544
## Water (component 4)(kg in a m^3 mixture) -0.08154361
## Superplasticizer (component 5)(kg in a m^3 mixture) 0.09277137
## Coarse Aggregate (component 6)(kg in a m^3 mixture) -0.10935604
## Fine Aggregate (component 7)(kg in a m^3 mixture) -0.22272017
## Age (day) 0.08194726
## Concrete compressive strength(MPa, megapascals) 0.49783272
## Blast Furnace Slag (component 2)(kg in a m^3 mixture)
## Cement (component 1)(kg in a m^3 mixture) -0.27519344
## Blast Furnace Slag (component 2)(kg in a m^3 mixture) 1.00000000
## Fly Ash (component 3)(kg in a m^3 mixture) -0.32356947
## Water (component 4)(kg in a m^3 mixture) 0.10728594
## Superplasticizer (component 5)(kg in a m^3 mixture) 0.04337574
## Coarse Aggregate (component 6)(kg in a m^3 mixture) -0.28399823
## Fine Aggregate (component 7)(kg in a m^3 mixture) -0.28159326
## Age (day) -0.04424580
## Concrete compressive strength(MPa, megapascals) 0.13482445
## Fly Ash (component 3)(kg in a m^3 mixture)
## Cement (component 1)(kg in a m^3 mixture) -0.397475440
## Blast Furnace Slag (component 2)(kg in a m^3 mixture) -0.323569468
## Fly Ash (component 3)(kg in a m^3 mixture) 1.000000000
## Water (component 4)(kg in a m^3 mixture) -0.257043997
## Superplasticizer (component 5)(kg in a m^3 mixture) 0.377339559
## Coarse Aggregate (component 6)(kg in a m^3 mixture) -0.009976788
## Fine Aggregate (component 7)(kg in a m^3 mixture) 0.079076351
## Age (day) -0.154370165
## Concrete compressive strength(MPa, megapascals) -0.105753348
## Water (component 4)(kg in a m^3 mixture)
## Cement (component 1)(kg in a m^3 mixture) -0.08154361
## Blast Furnace Slag (component 2)(kg in a m^3 mixture) 0.10728594
## Fly Ash (component 3)(kg in a m^3 mixture) -0.25704400
## Water (component 4)(kg in a m^3 mixture) 1.00000000
## Superplasticizer (component 5)(kg in a m^3 mixture) -0.65746444
## Coarse Aggregate (component 6)(kg in a m^3 mixture) -0.18231167
## Fine Aggregate (component 7)(kg in a m^3 mixture) -0.45063498
## Age (day) 0.27760443
## Concrete compressive strength(MPa, megapascals) -0.28961348
## Superplasticizer (component 5)(kg in a m^3 mixture)
## Cement (component 1)(kg in a m^3 mixture) 0.09277137
## Blast Furnace Slag (component 2)(kg in a m^3 mixture) 0.04337574
## Fly Ash (component 3)(kg in a m^3 mixture) 0.37733956
## Water (component 4)(kg in a m^3 mixture) -0.65746444
## Superplasticizer (component 5)(kg in a m^3 mixture) 1.00000000
## Coarse Aggregate (component 6)(kg in a m^3 mixture) -0.26630276
## Fine Aggregate (component 7)(kg in a m^3 mixture) 0.22250149
## Age (day) -0.19271652
## Concrete compressive strength(MPa, megapascals) 0.36610230
## Coarse Aggregate (component 6)(kg in a m^3 mixture)
## Cement (component 1)(kg in a m^3 mixture) -0.109356039
## Blast Furnace Slag (component 2)(kg in a m^3 mixture) -0.283998230
## Fly Ash (component 3)(kg in a m^3 mixture) -0.009976788
## Water (component 4)(kg in a m^3 mixture) -0.182311668
## Superplasticizer (component 5)(kg in a m^3 mixture) -0.266302755
## Coarse Aggregate (component 6)(kg in a m^3 mixture) 1.000000000
## Fine Aggregate (component 7)(kg in a m^3 mixture) -0.178505755
## Age (day) -0.003015507
## Concrete compressive strength(MPa, megapascals) -0.164927821
## Fine Aggregate (component 7)(kg in a m^3 mixture)
## Cement (component 1)(kg in a m^3 mixture) -0.22272017
## Blast Furnace Slag (component 2)(kg in a m^3 mixture) -0.28159326
## Fly Ash (component 3)(kg in a m^3 mixture) 0.07907635
## Water (component 4)(kg in a m^3 mixture) -0.45063498
## Superplasticizer (component 5)(kg in a m^3 mixture) 0.22250149
## Coarse Aggregate (component 6)(kg in a m^3 mixture) -0.17850575
## Fine Aggregate (component 7)(kg in a m^3 mixture) 1.00000000
## Age (day) -0.15609405
## Concrete compressive strength(MPa, megapascals) -0.16724896
## Age (day)
## Cement (component 1)(kg in a m^3 mixture) 0.081947264
## Blast Furnace Slag (component 2)(kg in a m^3 mixture) -0.044245801
## Fly Ash (component 3)(kg in a m^3 mixture) -0.154370165
## Water (component 4)(kg in a m^3 mixture) 0.277604429
## Superplasticizer (component 5)(kg in a m^3 mixture) -0.192716518
## Coarse Aggregate (component 6)(kg in a m^3 mixture) -0.003015507
## Fine Aggregate (component 7)(kg in a m^3 mixture) -0.156094049
## Age (day) 1.000000000
## Concrete compressive strength(MPa, megapascals) 0.328876976
## Concrete compressive strength(MPa, megapascals)
## Cement (component 1)(kg in a m^3 mixture) 0.4978327
## Blast Furnace Slag (component 2)(kg in a m^3 mixture) 0.1348244
## Fly Ash (component 3)(kg in a m^3 mixture) -0.1057533
## Water (component 4)(kg in a m^3 mixture) -0.2896135
## Superplasticizer (component 5)(kg in a m^3 mixture) 0.3661023
## Coarse Aggregate (component 6)(kg in a m^3 mixture) -0.1649278
## Fine Aggregate (component 7)(kg in a m^3 mixture) -0.1672490
## Age (day) 0.3288770
## Concrete compressive strength(MPa, megapascals) 1.0000000Matriks korelasi dihitung untuk mengetahui seberapa kuat hubungan antar variabel. Nilai korelasi berada di antara -1 sampai 1. Nilai yang mendekati 1 menunjukkan hubungan positif yang kuat, sedangkan nilai mendekati -1 menunjukkan hubungan negatif yang kuat. Jika nilainya mendekati 0, maka hubungan antar variabel cenderung lemah. Dari matriks ini dapat dilihat bahwa beberapa variabel campuran beton memiliki hubungan tertentu dengan kuat tekan beton.
Visualisasi Correlation Matrix
corrplot(cor_matrix,
method = "color",
type = "upper",
tl.cex = 0.7)
Visualisasi dilakukan menggunakan corrplot() agar
hubungan antar variabel lebih mudah dipahami. Warna yang semakin gelap
menunjukkan hubungan yang semakin kuat. Dengan grafik ini, pola hubungan
antar variabel dapat terlihat lebih jelas dibandingkan hanya membaca
tabel angka.
5. Variance-Covariance Matrix
cov_matrix <- cov(data_numeric)
cov_matrix## Cement (component 1)(kg in a m^3 mixture)
## Cement (component 1)(kg in a m^3 mixture) 10921.74265
## Blast Furnace Slag (component 2)(kg in a m^3 mixture) -2481.35943
## Fly Ash (component 3)(kg in a m^3 mixture) -2658.35075
## Water (component 4)(kg in a m^3 mixture) -181.98979
## Superplasticizer (component 5)(kg in a m^3 mixture) 57.91462
## Coarse Aggregate (component 6)(kg in a m^3 mixture) -888.60851
## Fine Aggregate (component 7)(kg in a m^3 mixture) -1866.15111
## Age (day) 540.99182
## Concrete compressive strength(MPa, megapascals) 869.14762
## Blast Furnace Slag (component 2)(kg in a m^3 mixture)
## Cement (component 1)(kg in a m^3 mixture) -2481.35943
## Blast Furnace Slag (component 2)(kg in a m^3 mixture) 7444.08373
## Fly Ash (component 3)(kg in a m^3 mixture) -1786.60759
## Water (component 4)(kg in a m^3 mixture) 197.67855
## Superplasticizer (component 5)(kg in a m^3 mixture) 22.35531
## Coarse Aggregate (component 6)(kg in a m^3 mixture) -1905.21057
## Fine Aggregate (component 7)(kg in a m^3 mixture) -1947.91126
## Age (day) -241.15038
## Concrete compressive strength(MPa, megapascals) 194.32935
## Fly Ash (component 3)(kg in a m^3 mixture)
## Cement (component 1)(kg in a m^3 mixture) -2658.3508
## Blast Furnace Slag (component 2)(kg in a m^3 mixture) -1786.6076
## Fly Ash (component 3)(kg in a m^3 mixture) 4095.5481
## Water (component 4)(kg in a m^3 mixture) -351.2971
## Superplasticizer (component 5)(kg in a m^3 mixture) 144.2503
## Coarse Aggregate (component 6)(kg in a m^3 mixture) -49.6442
## Fine Aggregate (component 7)(kg in a m^3 mixture) 405.7364
## Age (day) -624.0647
## Concrete compressive strength(MPa, megapascals) -113.0614
## Water (component 4)(kg in a m^3 mixture)
## Cement (component 1)(kg in a m^3 mixture) -181.98979
## Blast Furnace Slag (component 2)(kg in a m^3 mixture) 197.67855
## Fly Ash (component 3)(kg in a m^3 mixture) -351.29712
## Water (component 4)(kg in a m^3 mixture) 456.06024
## Superplasticizer (component 5)(kg in a m^3 mixture) -83.87096
## Coarse Aggregate (component 6)(kg in a m^3 mixture) -302.72431
## Fine Aggregate (component 7)(kg in a m^3 mixture) -771.57347
## Age (day) 374.49650
## Concrete compressive strength(MPa, megapascals) -103.32229
## Superplasticizer (component 5)(kg in a m^3 mixture)
## Cement (component 1)(kg in a m^3 mixture) 57.91462
## Blast Furnace Slag (component 2)(kg in a m^3 mixture) 22.35531
## Fly Ash (component 3)(kg in a m^3 mixture) 144.25026
## Water (component 4)(kg in a m^3 mixture) -83.87096
## Superplasticizer (component 5)(kg in a m^3 mixture) 35.68260
## Coarse Aggregate (component 6)(kg in a m^3 mixture) -123.68745
## Fine Aggregate (component 7)(kg in a m^3 mixture) 106.56203
## Age (day) -72.72060
## Concrete compressive strength(MPa, megapascals) 36.53380
## Coarse Aggregate (component 6)(kg in a m^3 mixture)
## Cement (component 1)(kg in a m^3 mixture) -888.60851
## Blast Furnace Slag (component 2)(kg in a m^3 mixture) -1905.21057
## Fly Ash (component 3)(kg in a m^3 mixture) -49.64420
## Water (component 4)(kg in a m^3 mixture) -302.72431
## Superplasticizer (component 5)(kg in a m^3 mixture) -123.68745
## Coarse Aggregate (component 6)(kg in a m^3 mixture) 6045.65623
## Fine Aggregate (component 7)(kg in a m^3 mixture) -1112.79516
## Age (day) -14.81127
## Concrete compressive strength(MPa, megapascals) -214.22975
## Fine Aggregate (component 7)(kg in a m^3 mixture)
## Cement (component 1)(kg in a m^3 mixture) -1866.1511
## Blast Furnace Slag (component 2)(kg in a m^3 mixture) -1947.9113
## Fly Ash (component 3)(kg in a m^3 mixture) 405.7364
## Water (component 4)(kg in a m^3 mixture) -771.5735
## Superplasticizer (component 5)(kg in a m^3 mixture) 106.5620
## Coarse Aggregate (component 6)(kg in a m^3 mixture) -1112.7952
## Fine Aggregate (component 7)(kg in a m^3 mixture) 6428.0992
## Age (day) -790.5656
## Concrete compressive strength(MPa, megapascals) -224.0107
## Age (day)
## Cement (component 1)(kg in a m^3 mixture) 540.99182
## Blast Furnace Slag (component 2)(kg in a m^3 mixture) -241.15038
## Fly Ash (component 3)(kg in a m^3 mixture) -624.06475
## Water (component 4)(kg in a m^3 mixture) 374.49650
## Superplasticizer (component 5)(kg in a m^3 mixture) -72.72060
## Coarse Aggregate (component 6)(kg in a m^3 mixture) -14.81127
## Fine Aggregate (component 7)(kg in a m^3 mixture) -790.56558
## Age (day) 3990.43773
## Concrete compressive strength(MPa, megapascals) 347.06265
## Concrete compressive strength(MPa, megapascals)
## Cement (component 1)(kg in a m^3 mixture) 869.1476
## Blast Furnace Slag (component 2)(kg in a m^3 mixture) 194.3294
## Fly Ash (component 3)(kg in a m^3 mixture) -113.0614
## Water (component 4)(kg in a m^3 mixture) -103.3223
## Superplasticizer (component 5)(kg in a m^3 mixture) 36.5338
## Coarse Aggregate (component 6)(kg in a m^3 mixture) -214.2298
## Fine Aggregate (component 7)(kg in a m^3 mixture) -224.0107
## Age (day) 347.0626
## Concrete compressive strength(MPa, megapascals) 279.0797Matriks kovarians dihitung menggunakan fungsi cov().
Nilai pada diagonal menunjukkan varians dari masing-masing variabel,
sedangkan nilai di luar diagonal menunjukkan kovarians antar dua
variabel. Kovarians positif berarti kedua variabel bergerak searah,
sedangkan kovarians negatif berarti bergerak berlawanan arah. Namun
karena setiap variabel memiliki satuan yang berbeda, besar kecilnya
kovarians tidak selalu mudah dibandingkan secara langsung.
6. Eigen Value dan Eigen Vector
eigen_result <- eigen(cor_matrix)
eigen_values <- eigen_result$values
eigen_vectors <- eigen_result$vectors
eigen_values## [1] 2.28771185 1.93651535 1.40892580 1.04278807 1.01415431 0.84741063 0.28695777
## [8] 0.14678093 0.02875528eigen_vectors## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0.04106772 0.5364728 0.359699156 -0.30976550 -0.0546888297 -0.3898607
## [2,] 0.16299258 0.1363006 -0.698988731 0.07626221 -0.3625936171 0.2703252
## [3,] -0.36981124 -0.2684064 0.019777961 0.60066828 0.2275943912 -0.3202302
## [4,] 0.56408457 -0.1181279 -0.120302488 0.04692072 0.2960859177 -0.3061956
## [5,] -0.53605897 0.2482312 -0.187959037 0.16585735 -0.0369894532 -0.0827810
## [6,] 0.06045806 -0.2248310 0.549499919 0.22164550 -0.5454645518 0.3475894
## [7,] -0.38168340 -0.1870620 0.001233946 -0.52781995 0.3844845216 0.4091099
## [8,] 0.26191454 0.2518295 0.169597266 0.35951724 0.5285269011 0.5097793
## [9,] -0.10723462 0.6301150 0.033524610 0.22526326 0.0003062718 0.1539897
## [,7] [,8] [,9]
## [1,] -0.133772604 -0.2983673 -0.47251654
## [2,] 0.004823084 -0.2287723 -0.45115907
## [3,] 0.247179667 -0.2553441 -0.38647003
## [4,] -0.009807539 0.5855720 -0.35604330
## [5,] -0.613878507 0.4475792 -0.05281921
## [6,] -0.059842857 0.2430661 -0.33720421
## [7,] 0.174686900 0.1403404 -0.41871225
## [8,] -0.343644186 -0.2260221 -0.03967220
## [9,] 0.625981489 0.3469226 0.06055837Eigen value dan eigen vector dihitung dari matriks korelasi. Eigen value menunjukkan seberapa besar variasi data yang dapat dijelaskan oleh masing-masing komponen utama. Semakin besar nilai eigen, maka semakin besar kontribusi komponen tersebut.
Sedangkan eigen vector menunjukkan bagaimana masing-masing variabel berkontribusi dalam membentuk komponen utama tersebut.
7. Proporsi Variasi
proporsi_variasi <- eigen_values / sum(eigen_values)
proporsi_variasi## [1] 0.254190206 0.215168373 0.156547311 0.115865342 0.112683812 0.094156736
## [7] 0.031884197 0.016308992 0.003195031Proporsi variasi dihitung untuk mengetahui persentase kontribusi setiap komponen utama terhadap total variasi data. Dari hasil ini dapat dilihat komponen mana yang paling dominan dalam menjelaskan variasi data secara keseluruhan.
8. Scree Plot
plot(eigen_values,
type="b",
main="Scree Plot",
xlab="Komponen",
ylab="Eigen Value")
Scree plot dibuat untuk membantu menentukan jumlah komponen utama yang sebaiknya dipertahankan. Komponen dengan eigen value lebih besar dari 1 biasanya dianggap cukup penting. Grafik ini juga membantu melihat titik di mana penurunan nilai eigen mulai melandai.
9. Kesimpulan
Berdasarkan hasil analisis, terlihat bahwa terdapat hubungan antar beberapa variabel dalam campuran beton terhadap kuat tekan beton. Matriks korelasi menunjukkan adanya hubungan positif dan negatif antar variabel. Dari hasil eigen value, komponen utama pertama memiliki kontribusi variasi terbesar dibandingkan komponen lainnya. Dengan menggunakan beberapa komponen utama saja, sebagian besar informasi dalam data sudah dapat dijelaskan tanpa harus menggunakan seluruh variabel asli.