1 Pendahuluan

Analisis ini bertujuan untuk mengeksplorasi hubungan antar variabel dalam Concrete Compressive Strength Dataset menggunakan teknik analisis multivariat:

  • Correlation Matrix: Mengukur kekuatan hubungan linear antar variabel
  • Variance-Covariance Matrix: Mengukur variabilitas dan kovarians
  • Eigenvalue & Eigenvector: Ekstraksi principal components

2 Import dan Eksplorasi Data

2.1 Load Libraries

library(readxl)
library(corrplot)
library(ggplot2)
library(knitr)
library(kableExtra)
library(dplyr)

2.2 Import Dataset

data_raw <- read_excel("Concrete_Data.xls")
data <- data_raw %>% select(where(is.numeric))
data <- na.omit(data)

2.3 Preview Data

head(data, 10) %>%
  kable(caption = "10 Observasi Pertama", digits = 2) %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"), 
                full_width = FALSE,
                position = "center")
10 Observasi Pertama
Cement (component 1)(kg in a m^3 mixture) Blast Furnace Slag (component 2)(kg in a m^3 mixture) Fly Ash (component 3)(kg in a m^3 mixture) Water (component 4)(kg in a m^3 mixture) Superplasticizer (component 5)(kg in a m^3 mixture) Coarse Aggregate (component 6)(kg in a m^3 mixture) Fine Aggregate (component 7)(kg in a m^3 mixture) Age (day) Concrete compressive strength(MPa, megapascals)
540.0 0.0 0 162 2.5 1040.0 676.0 28 79.99
540.0 0.0 0 162 2.5 1055.0 676.0 28 61.89
332.5 142.5 0 228 0.0 932.0 594.0 270 40.27
332.5 142.5 0 228 0.0 932.0 594.0 365 41.05
198.6 132.4 0 192 0.0 978.4 825.5 360 44.30
266.0 114.0 0 228 0.0 932.0 670.0 90 47.03
380.0 95.0 0 228 0.0 932.0 594.0 365 43.70
380.0 95.0 0 228 0.0 932.0 594.0 28 36.45
266.0 114.0 0 228 0.0 932.0 670.0 28 45.85
475.0 0.0 0 228 0.0 932.0 594.0 28 39.29

2.4 Statistik Deskriptif

summary(data) %>%
  kable(caption = "Statistik Deskriptif", digits = 2) %>%
  kable_styling(bootstrap_options = c("striped", "hover"), 
                full_width = FALSE,
                position = "center")
Statistik Deskriptif
Cement (component 1)(kg in a m^3 mixture) Blast Furnace Slag (component 2)(kg in a m^3 mixture) Fly Ash (component 3)(kg in a m^3 mixture) Water (component 4)(kg in a m^3 mixture) Superplasticizer (component 5)(kg in a m^3 mixture) Coarse Aggregate (component 6)(kg in a m^3 mixture) Fine Aggregate (component 7)(kg in a m^3 mixture) Age (day) Concrete compressive strength(MPa, megapascals)
Min. :102.0 Min. : 0.0 Min. : 0.00 Min. :121.8 Min. : 0.000 Min. : 801.0 Min. :594.0 Min. : 1.00 Min. : 2.332
1st Qu.:192.4 1st Qu.: 0.0 1st Qu.: 0.00 1st Qu.:164.9 1st Qu.: 0.000 1st Qu.: 932.0 1st Qu.:731.0 1st Qu.: 7.00 1st Qu.:23.707
Median :272.9 Median : 22.0 Median : 0.00 Median :185.0 Median : 6.350 Median : 968.0 Median :779.5 Median : 28.00 Median :34.443
Mean :281.2 Mean : 73.9 Mean : 54.19 Mean :181.6 Mean : 6.203 Mean : 972.9 Mean :773.6 Mean : 45.66 Mean :35.818
3rd Qu.:350.0 3rd Qu.:142.9 3rd Qu.:118.27 3rd Qu.:192.0 3rd Qu.:10.160 3rd Qu.:1029.4 3rd Qu.:824.0 3rd Qu.: 56.00 3rd Qu.:46.136
Max. :540.0 Max. :359.4 Max. :200.10 Max. :247.0 Max. :32.200 Max. :1145.0 Max. :992.6 Max. :365.00 Max. :82.599

Dimensi Data: 1030 observasi × 9 variabel

3 Correlation Matrix

3.1 Perhitungan

Correlation matrix mengukur kekuatan dan arah hubungan linear antara pasangan variabel dengan nilai berkisar antara -1 hingga +1.

Formula: \[r_{xy} = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum(x_i - \bar{x})^2 \sum(y_i - \bar{y})^2}}\]

corr_matrix <- cor(data)

corr_matrix %>%
  kable(caption = "Correlation Matrix", digits = 4) %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"),
                font_size = 11) %>%
  scroll_box(width = "100%", height = "400px")
Correlation Matrix
Cement (component 1)(kg in a m^3 mixture) Blast Furnace Slag (component 2)(kg in a m^3 mixture) Fly Ash (component 3)(kg in a m^3 mixture) Water (component 4)(kg in a m^3 mixture) Superplasticizer (component 5)(kg in a m^3 mixture) Coarse Aggregate (component 6)(kg in a m^3 mixture) Fine Aggregate (component 7)(kg in a m^3 mixture) Age (day) Concrete compressive strength(MPa, megapascals)
Cement (component 1)(kg in a m^3 mixture) 1.0000 -0.2752 -0.3975 -0.0815 0.0928 -0.1094 -0.2227 0.0819 0.4978
Blast Furnace Slag (component 2)(kg in a m^3 mixture) -0.2752 1.0000 -0.3236 0.1073 0.0434 -0.2840 -0.2816 -0.0442 0.1348
Fly Ash (component 3)(kg in a m^3 mixture) -0.3975 -0.3236 1.0000 -0.2570 0.3773 -0.0100 0.0791 -0.1544 -0.1058
Water (component 4)(kg in a m^3 mixture) -0.0815 0.1073 -0.2570 1.0000 -0.6575 -0.1823 -0.4506 0.2776 -0.2896
Superplasticizer (component 5)(kg in a m^3 mixture) 0.0928 0.0434 0.3773 -0.6575 1.0000 -0.2663 0.2225 -0.1927 0.3661
Coarse Aggregate (component 6)(kg in a m^3 mixture) -0.1094 -0.2840 -0.0100 -0.1823 -0.2663 1.0000 -0.1785 -0.0030 -0.1649
Fine Aggregate (component 7)(kg in a m^3 mixture) -0.2227 -0.2816 0.0791 -0.4506 0.2225 -0.1785 1.0000 -0.1561 -0.1672
Age (day) 0.0819 -0.0442 -0.1544 0.2776 -0.1927 -0.0030 -0.1561 1.0000 0.3289
Concrete compressive strength(MPa, megapascals) 0.4978 0.1348 -0.1058 -0.2896 0.3661 -0.1649 -0.1672 0.3289 1.0000

3.2 Visualisasi

corrplot(corr_matrix, 
         method = "color",
         type = "upper",
         order = "hclust",
         addCoef.col = "black",
         tl.col = "black",
         tl.srt = 45,
         number.cex = 0.6,
         tl.cex = 0.8,
         col = colorRampPalette(c("#6D9EC1", "white", "#E46726"))(200),
         title = "Correlation Matrix - Concrete Data",
         mar = c(0,0,2,0))
Heatmap Correlation Matrix

Heatmap Correlation Matrix

3.3 Interpretasi

Kriteria kekuatan korelasi:

Nilai |r| Interpretasi
0.00 - 0.19 Sangat Lemah
0.20 - 0.39 Lemah
0.40 - 0.59 Sedang
0.60 - 0.79 Kuat
0.80 - 1.00 Sangat Kuat 
cor_melted <- data.frame(
  Var1 = rownames(corr_matrix)[row(corr_matrix)],
  Var2 = colnames(corr_matrix)[col(corr_matrix)],
  Correlation = c(corr_matrix)
) %>%
  filter(Var1 != Var2) %>%
  mutate(Abs_Cor = abs(Correlation)) %>%
  arrange(desc(Abs_Cor)) %>%
  head(5)

cor_melted %>%
  select(Var1, Var2, Correlation) %>%
  kable(caption = "Top 5 Korelasi Tertinggi", 
        digits = 4,
        col.names = c("Variabel 1", "Variabel 2", "Korelasi")) %>%
  kable_styling(bootstrap_options = c("striped", "hover"),
                full_width = FALSE,
                position = "center")
Top 5 Korelasi Tertinggi
Variabel 1 Variabel 2 Korelasi
Superplasticizer (component 5)(kg in a m^3 mixture) Water (component 4)(kg in a m^3 mixture) -0.6575
Water (component 4)(kg in a m^3 mixture) Superplasticizer (component 5)(kg in a m^3 mixture) -0.6575
Concrete compressive strength(MPa, megapascals) Cement (component 1)(kg in a m^3 mixture) 0.4978
Cement (component 1)(kg in a m^3 mixture) Concrete compressive strength(MPa, megapascals) 0.4978
Fine Aggregate (component 7)(kg in a m^3 mixture) Water (component 4)(kg in a m^3 mixture) -0.4506

4 Variance-Covariance Matrix

4.1 Perhitungan

Variance-Covariance matrix menunjukkan:

  • Diagonal: Variance (keragaman) masing-masing variabel
  • Off-diagonal: Covariance (hubungan variabilitas) antar variabel

Formula:

Variance: \(\sigma^2_x = \frac{\sum(x_i - \bar{x})^2}{n-1}\)

Covariance: \(Cov(x,y) = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{n-1}\)

cov_matrix <- cov(data)

cov_matrix %>%
  kable(caption = "Variance-Covariance Matrix", digits = 2) %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"),
                font_size = 11) %>%
  scroll_box(width = "100%", height = "400px")
Variance-Covariance Matrix
Cement (component 1)(kg in a m^3 mixture) Blast Furnace Slag (component 2)(kg in a m^3 mixture) Fly Ash (component 3)(kg in a m^3 mixture) Water (component 4)(kg in a m^3 mixture) Superplasticizer (component 5)(kg in a m^3 mixture) Coarse Aggregate (component 6)(kg in a m^3 mixture) Fine Aggregate (component 7)(kg in a m^3 mixture) Age (day) Concrete compressive strength(MPa, megapascals)
Cement (component 1)(kg in a m^3 mixture) 10921.74 -2481.36 -2658.35 -181.99 57.91 -888.61 -1866.15 540.99 869.15
Blast Furnace Slag (component 2)(kg in a m^3 mixture) -2481.36 7444.08 -1786.61 197.68 22.36 -1905.21 -1947.91 -241.15 194.33
Fly Ash (component 3)(kg in a m^3 mixture) -2658.35 -1786.61 4095.55 -351.30 144.25 -49.64 405.74 -624.06 -113.06
Water (component 4)(kg in a m^3 mixture) -181.99 197.68 -351.30 456.06 -83.87 -302.72 -771.57 374.50 -103.32
Superplasticizer (component 5)(kg in a m^3 mixture) 57.91 22.36 144.25 -83.87 35.68 -123.69 106.56 -72.72 36.53
Coarse Aggregate (component 6)(kg in a m^3 mixture) -888.61 -1905.21 -49.64 -302.72 -123.69 6045.66 -1112.80 -14.81 -214.23
Fine Aggregate (component 7)(kg in a m^3 mixture) -1866.15 -1947.91 405.74 -771.57 106.56 -1112.80 6428.10 -790.57 -224.01
Age (day) 540.99 -241.15 -624.06 374.50 -72.72 -14.81 -790.57 3990.44 347.06
Concrete compressive strength(MPa, megapascals) 869.15 194.33 -113.06 -103.32 36.53 -214.23 -224.01 347.06 279.08

4.2 Variance per Variabel

variances <- diag(cov_matrix)

variance_table <- data.frame(
  Variabel = names(variances),
  Variance = variances,
  Std_Dev = sqrt(variances)
) %>%
  arrange(desc(Variance))

variance_table %>%
  kable(caption = "Variance dan Standar Deviasi", 
        digits = 2,
        col.names = c("Variabel", "Variance", "Standar Deviasi")) %>%
  kable_styling(bootstrap_options = c("striped", "hover"),
                full_width = FALSE,
                position = "center")
Variance dan Standar Deviasi
Variabel Variance Standar Deviasi
Cement (component 1)(kg in a m^3 mixture) Cement (component 1)(kg in a m^3 mixture) 10921.74 104.51
Blast Furnace Slag (component 2)(kg in a m^3 mixture) Blast Furnace Slag (component 2)(kg in a m^3 mixture) 7444.08 86.28
Fine Aggregate (component 7)(kg in a m^3 mixture) Fine Aggregate (component 7)(kg in a m^3 mixture) 6428.10 80.18
Coarse Aggregate (component 6)(kg in a m^3 mixture) Coarse Aggregate (component 6)(kg in a m^3 mixture) 6045.66 77.75
Fly Ash (component 3)(kg in a m^3 mixture) Fly Ash (component 3)(kg in a m^3 mixture) 4095.55 64.00
Age (day) Age (day) 3990.44 63.17
Water (component 4)(kg in a m^3 mixture) Water (component 4)(kg in a m^3 mixture) 456.06 21.36
Concrete compressive strength(MPa, megapascals) Concrete compressive strength(MPa, megapascals) 279.08 16.71
Superplasticizer (component 5)(kg in a m^3 mixture) Superplasticizer (component 5)(kg in a m^3 mixture) 35.68 5.97 
ggplot(variance_table, aes(x = reorder(Variabel, Variance), y = Variance)) +
  geom_bar(stat = "identity", fill = "#3498db", alpha = 0.8) +
  coord_flip() +
  theme_minimal() +
  labs(title = "Variance per Variabel",
       x = "",
       y = "Variance") +
  theme(
    plot.title = element_text(hjust = 0.5, face = "bold", size = 14),
    axis.text = element_text(size = 10)
  )
Variance per Variabel

Variance per Variabel

4.3 Interpretasi

  • Covariance positif: Kedua variabel cenderung bergerak searah
  • Covariance negatif: Kedua variabel cenderung bergerak berlawanan
  • Variabel dengan variance tertinggi memiliki variabilitas data terbesar

5 Eigenvalue dan Eigenvector

5.1 Dari Correlation Matrix

5.1.1 Eigenvalues

eigen_result <- eigen(corr_matrix)
prop_var <- eigen_result$values / sum(eigen_result$values)
cumsum_var <- cumsum(prop_var)

eigen_table <- data.frame(
  PC = paste0("PC", 1:length(eigen_result$values)),
  Eigenvalue = eigen_result$values,
  Proportion = prop_var * 100,
  Cumulative = cumsum_var * 100
)

eigen_table %>%
  kable(caption = "Eigenvalue Summary", 
        digits = 4,
        col.names = c("PC", "Eigenvalue", "Proporsi (%)", "Kumulatif (%)")) %>%
  kable_styling(bootstrap_options = c("striped", "hover"),
                full_width = FALSE,
                position = "center")
Eigenvalue Summary
PC Eigenvalue Proporsi (%) Kumulatif (%)
PC1 2.2877 25.4190 25.4190
PC2 1.9365 21.5168 46.9359
PC3 1.4089 15.6547 62.5906
PC4 1.0428 11.5865 74.1771
PC5 1.0142 11.2684 85.4455
PC6 0.8474 9.4157 94.8612
PC7 0.2870 3.1884 98.0496
PC8 0.1468 1.6309 99.6805
PC9 0.0288 0.3195 100.0000

5.1.2 Scree Plot

ggplot(eigen_table, aes(x = PC, y = Eigenvalue, group = 1)) +
  geom_line(color = "#3498db", size = 1.2) +
  geom_point(color = "#e74c3c", size = 3) +
  geom_hline(yintercept = 1, linetype = "dashed", color = "#27ae60", size = 1) +
  annotate("text", x = nrow(eigen_table) * 0.7, y = 1.2, 
           label = "Kaiser Criterion (λ = 1)", color = "#27ae60") +
  theme_minimal() +
  labs(
    title = "Scree Plot",
    subtitle = "Eigenvalue dari Correlation Matrix",
    x = "Principal Component",
    y = "Eigenvalue"
  ) +
  theme(
    plot.title = element_text(hjust = 0.5, face = "bold", size = 14),
    plot.subtitle = element_text(hjust = 0.5, size = 11),
    axis.text.x = element_text(angle = 45, hjust = 1)
  )
Scree Plot - Eigenvalue vs Principal Component

Scree Plot - Eigenvalue vs Principal Component

5.1.3 Eigenvectors

eigen_result$vectors %>%
  as.data.frame() %>%
  setNames(paste0("PC", 1:ncol(.))) %>%
  mutate(Variabel = names(data)) %>%
  select(Variabel, everything()) %>%
  kable(caption = "Eigenvectors (Loading Matrix)", digits = 4) %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"),
                font_size = 10) %>%
  scroll_box(width = "100%", height = "400px")
Eigenvectors (Loading Matrix)
Variabel PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8 PC9
Cement (component 1)(kg in a m^3 mixture) -0.0411 0.5365 0.3597 -0.3098 -0.0547 -0.3899 -0.1338 -0.2984 -0.4725
Blast Furnace Slag (component 2)(kg in a m^3 mixture) -0.1630 0.1363 -0.6990 0.0763 -0.3626 0.2703 0.0048 -0.2288 -0.4512
Fly Ash (component 3)(kg in a m^3 mixture) 0.3698 -0.2684 0.0198 0.6007 0.2276 -0.3202 0.2472 -0.2553 -0.3865
Water (component 4)(kg in a m^3 mixture) -0.5641 -0.1181 -0.1203 0.0469 0.2961 -0.3062 -0.0098 0.5856 -0.3560
Superplasticizer (component 5)(kg in a m^3 mixture) 0.5361 0.2482 -0.1880 0.1659 -0.0370 -0.0828 -0.6139 0.4476 -0.0528
Coarse Aggregate (component 6)(kg in a m^3 mixture) -0.0605 -0.2248 0.5495 0.2216 -0.5455 0.3476 -0.0598 0.2431 -0.3372
Fine Aggregate (component 7)(kg in a m^3 mixture) 0.3817 -0.1871 0.0012 -0.5278 0.3845 0.4091 0.1747 0.1403 -0.4187
Age (day) -0.2619 0.2518 0.1696 0.3595 0.5285 0.5098 -0.3436 -0.2260 -0.0397
Concrete compressive strength(MPa, megapascals) 0.1072 0.6301 0.0335 0.2253 0.0003 0.1540 0.6260 0.3469 0.0606

5.2 Interpretasi

5.2.1 Definisi Matematika

Eigenvalue & Eigenvector memenuhi persamaan: \[A \mathbf{v} = \lambda \mathbf{v}\]

Dimana:

  • \(A\) = Correlation/Covariance matrix
  • \(\mathbf{v}\) = Eigenvector
  • \(\lambda\) = Eigenvalue

5.2.2 Temuan

n_significant <- sum(eigen_result$values > 1)
  • PC1 menjelaskan 25.42% dari total varians
  • PC2 menjelaskan 21.52% dari total varians
  • PC1 & PC2 secara kumulatif: 46.94%
  • Berdasarkan Kaiser Criterion (eigenvalue > 1): 5 PC signifikan
  • 5 PC menjelaskan 85.45% total varians

5.2.3 Kesimpulan

Dimensi data dapat direduksi dari 9 variabel menjadi 5 Principal Components dengan tetap mempertahankan 85.45% informasi.

6 Verifikasi Matematika

Memverifikasi sifat eigenvalue dan eigenvector: \(A \cdot \mathbf{v} = \lambda \cdot \mathbf{v}\)

lambda1 <- eigen_result$values[1]
v1 <- eigen_result$vectors[, 1]

Av <- corr_matrix %*% v1
lambda_v <- lambda1 * v1

comparison <- data.frame(
  Komponen = paste0("v", 1:length(v1)),
  Av = as.vector(Av),
  Lambda_v = as.vector(lambda_v),
  Difference = as.vector(Av - lambda_v)
)

comparison %>%
  kable(caption = "Verifikasi: A·v vs λ·v", 
        digits = 8,
        col.names = c("Komponen", "A·v", "λ·v", "Selisih")) %>%
  kable_styling(bootstrap_options = c("striped", "hover"),
                full_width = FALSE,
                position = "center")
Verifikasi: A·v vs λ·v
Komponen A·v λ·v Selisih
v1 -0.09395112 -0.09395112 0
v2 -0.37288006 -0.37288006 0
v3 0.84602156 0.84602156 0
v4 -1.29046295 -1.29046295 0
v5 1.22634847 1.22634847 0
v6 -0.13831061 -0.13831061 0
v7 0.87318163 0.87318163 0
v8 -0.59918501 -0.59918501 0
v9 0.24532192 0.24532192 0

Maksimum selisih: 2.89e-15

✅ Verifikasi berhasil (nilai mendekati 0)

7 Kesimpulan

Dari analisis Concrete Compressive Strength Dataset, dapat disimpulkan:

  1. Correlation Matrix menunjukkan hubungan antar variabel dalam bentuk standardized, memudahkan perbandingan kekuatan hubungan

  2. Variance-Covariance Matrix memberikan informasi tentang variabilitas masing-masing variabel dan kovarians antar variabel

  3. Principal Component Analysis menunjukkan bahwa dimensi data dapat direduksi secara signifikan:

    • Dari 9 variabel → 5 PC
    • Mempertahankan 85.45% informasi

  4. Hasil ini dapat digunakan untuk:

    • Reduksi dimensi dalam modeling
    • Visualisasi data multidimensi
    • Feature extraction
    • Identifikasi pola dalam data concrete

8 Referensi

  • Hair, J.F. (2018). Multivariate Data Analysis, 8th Ed. Cengage
  • Härdle, W.K., & Simar, L. (2019). Applied Multivariate Statistical Analysis. Springer Nature
  • Materi Kuliah Analisis Multivariat, Prodi S1 Sains Data, Universitas Negeri Surabaya

Tugas Analisis Multivariat

Review Matrix Algebra - Concrete Data Analysis

Program Studi S1 Sains Data

Fakultas Matematika dan Ilmu Pengetahuan Alam

Universitas Negeri Surabaya

2026-02-10