import data
library(readxl)
data <- read_excel("Concrete_Data.xls")
df <- as.data.frame(data)
class(df)## [1] "data.frame"str(df)## 'data.frame': 1030 obs. of 9 variables:
## $ Cement (component 1)(kg in a m^3 mixture) : num 540 540 332 332 199 ...
## $ Blast Furnace Slag (component 2)(kg in a m^3 mixture): num 0 0 142 142 132 ...
## $ Fly Ash (component 3)(kg in a m^3 mixture) : num 0 0 0 0 0 0 0 0 0 0 ...
## $ Water (component 4)(kg in a m^3 mixture) : num 162 162 228 228 192 228 228 228 228 228 ...
## $ Superplasticizer (component 5)(kg in a m^3 mixture) : num 2.5 2.5 0 0 0 0 0 0 0 0 ...
## $ Coarse Aggregate (component 6)(kg in a m^3 mixture) : num 1040 1055 932 932 978 ...
## $ Fine Aggregate (component 7)(kg in a m^3 mixture) : num 676 676 594 594 826 ...
## $ Age (day) : num 28 28 270 365 360 90 365 28 28 28 ...
## $ Concrete compressive strength(MPa, megapascals) : num 80 61.9 40.3 41.1 44.3 ...A. Korelasi matrix
cor_matrix <- cor(data)
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) 1.0000000Penjelasan : Matriks korelasi menunjukkan hubungan antar variabel pada data beton. Kekuatan tekan beton (Concrete compressive strength) memiliki korelasi positif sedang dengan Cement (0,498), Superplasticizer (0,366), dan Age (0,329), yang berarti semakin tinggi nilai variabel-variabel tersebut, kekuatan beton cenderung meningkat. Sebaliknya, kekuatan beton berkorelasi negatif dengan Water (-0,290), Coarse Aggregate (-0,165), dan Fine Aggregate (-0,167), yang menunjukkan bahwa penambahan air dan agregat berlebih cenderung menurunkan kekuatan beton. Hubungan antar variabel lain umumnya lemah hingga sedang.
B. Varians dan Kovarians matrix
cov_matrix <- cov(data)
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) 279.0797Penjelasan : Matriks varians–kovarians menunjukkan besar penyebaran data dan hubungan perubahan antar variabel. Nilai diagonal utama merupakan varians, misalnya Cement (10921,74) dan Age (3990,44) memiliki variasi paling besar, artinya data kedua variabel ini paling menyebar. Nilai di luar diagonal adalah kovarians, yang menunjukkan arah hubungan. Kovarians positif (misalnya Cement dengan Concrete strength = 869,15) berarti keduanya cenderung meningkat bersama, sedangkan kovarians negatif (misalnya Water dengan Concrete strength = -103,32) menunjukkan bahwa peningkatan air cenderung menurunkan kekuatan beton. Matriks ini menjadi dasar untuk analisis lanjutan seperti PCA.
C. Eigen Value dan Eigen Vektor
eigen_result <- eigen(cov_matrix)
eigen_result$values## [1] 12897.94299 9825.43415 7287.26315 4247.63405 3986.92184 1268.12213
## [7] 102.07298 69.74592 11.25295eigen_result$vectors## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0.904445813 -0.023040685 -0.15203754 0.01346216 0.15375901 -0.2767427
## [2,] -0.254633567 -0.788837027 -0.07143657 0.20066782 0.10135129 -0.4338754
## [3,] -0.239384888 0.299039739 0.04888714 -0.68625730 0.18776084 -0.4954283
## [4,] 0.005428255 -0.075493692 0.04205631 -0.07576154 -0.09397864 0.4679254
## [5,] -0.001101790 0.004857144 -0.02419471 -0.02038522 0.02279838 -0.1013839
## [6,] -0.012822425 0.276099110 0.75984686 0.47859604 0.06207670 -0.2752859
## [7,] -0.212270079 0.446453833 -0.61270296 0.48052792 -0.14558390 -0.2556211
## [8,] 0.100313043 -0.069996987 0.11771897 -0.14656970 -0.94573844 -0.2043009
## [9,] 0.067216106 -0.040078608 -0.02018436 -0.03166726 -0.04484212 -0.2786040
## [,7] [,8] [,9]
## [1,] -0.18387377 0.1548744 -0.011235218
## [2,] -0.18288729 0.1881842 -0.012089533
## [3,] -0.19442343 0.2478254 0.003146049
## [4,] -0.07068464 0.8325592 -0.246628008
## [5,] 0.05590572 -0.2224131 -0.967255259
## [6,] -0.07624081 0.1732152 -0.041610727
## [7,] -0.10237149 0.2270640 -0.027084378
## [8,] -0.11288137 -0.0281684 -0.001245791
## [9,] 0.92616718 0.2328213 0.029035081Penjelasan : Nilai eigen menunjukkan besar variasi data yang dijelaskan oleh setiap komponen utama. Komponen pertama memiliki eigen value terbesar (12897,94), sehingga merupakan komponen paling penting karena menjelaskan variasi data terbesar, diikuti oleh komponen kedua dan ketiga. Semakin kecil eigen value, semakin kecil kontribusinya terhadap variasi data.
Eigen vector menunjukkan kontribusi masing-masing variabel dalam membentuk setiap komponen utama. Pada komponen pertama, variabel Cement memiliki bobot paling besar (0,90), sehingga paling berpengaruh dalam menjelaskan variasi utama data. Variabel lain memiliki kontribusi lebih kecil. Artinya, variasi terbesar pada data beton terutama dipengaruhi oleh jumlah semen, sedangkan komponen-komponen berikutnya dibentuk oleh kombinasi variabel lain.