Titanic Dataset Analysis: Correlation Matrix, Variance-Covariance Matrix, and Eigenvalues

1.1 Variable Used

We will analyze four numerical variables from the Titanic dataset:

1. Age: Passenger’s age in years
2. SibSp: Number of siblings/spouses aboard
3. Parch: Number of parents/children aboard
4. Fare: Ticket price in British pounds

These variables represent key demographic and economic characteristics that might relate to passenger survival patterns.

2.1 Load Library

library(tidyverse)
library(corrplot)
library(knitr)
library(DT)

2.2 Load the Dataset

titanic <- read.csv("Titanic-Dataset.csv")

cat("Dataset dimension:", dim(titanic)[1], "row x", dim(titanic)[2], "columns\n")

## Dataset dimension: 891 row x 12 columns

2.3 Dataset Preview

head(titanic) %>% 
  kable(caption = "Head Rows of the Titanic Dataset") %>%
  kableExtra::kable_styling(bootstrap_options = c("striped", "hover", "condensed"))

2.4 Variable Selection

Select only the 4 variables we need.

titanic_selected <- titanic %>%
  select(Age, SibSp, Parch, Fare)

head(titanic_selected)

##   Age SibSp Parch    Fare
## 1  22     1     0  7.2500
## 2  38     1     0 71.2833
## 3  26     0     0  7.9250
## 4  35     1     0 53.1000
## 5  35     0     0  8.0500
## 6  NA     0     0  8.4583

2.5 Handling Missing Values

titanic_clean <- titanic_selected %>%
  na.omit()

cat("Number of rows before removing NA:", nrow(titanic_selected), "\n")

## Number of rows before removing NA: 891

cat("Number of rows after removing NA:", nrow(titanic_clean), "\n")

## Number of rows after removing NA: 714

cat("Rows removed:", nrow(titanic_selected) - nrow(titanic_clean), "\n")

## Rows removed: 177

3.1 Descriptive Statistics

summary(titanic_clean) %>%
  kable(caption = "Descriptive Statistics") %>%
  kableExtra::kable_styling(bootstrap_options = c("striped", "hover"))

• Min/Max: The smallest and largest values
• Median: The middle value when data is sorted
• Mean: The average value
• 1st/3rd Quartile: 25% and 75% points in the data

par(mfrow = c(2, 2))

hist(titanic_clean$Age, 
     breaks = 30,
     col = "#3498db",
     border = "white",
     main = "Distribution of Age",
     xlab = "Age (years)",
     ylab = "Frequency",
     cex.main = 1.3)
abline(v = mean(titanic_clean$Age), col = "red", lwd = 2, lty = 2)
abline(v = median(titanic_clean$Age), col = "orange", lwd = 2, lty = 2)
legend("topright", 
       legend = c("Mean", "Median"), 
       col = c("red", "orange"), 
       lty = 2, lwd = 2,
       cex = 0.8)

hist(titanic_clean$SibSp, 
     breaks = seq(-0.5, max(titanic_clean$SibSp) + 0.5, 1),
     col = "#e74c3c",
     border = "white",
     main = "Distribution of Siblings/Spouses",
     xlab = "Number of Siblings/Spouses",
     ylab = "Frequency",
     cex.main = 1.3)

hist(titanic_clean$Parch, 
     breaks = seq(-0.5, max(titanic_clean$Parch) + 0.5, 1),
     col = "#2ecc71",
     border = "white",
     main = "Distribution of Parents/Children",
     xlab = "Number of Parents/Children",
     ylab = "Frequency",
     cex.main = 1.3)

hist(titanic_clean$Fare, 
     breaks = 50,
     col = "#f39c12",
     border = "white",
     main = "Distribution of Fare",
     xlab = "Fare (British Pounds)",
     ylab = "Frequency",
     cex.main = 1.3)
abline(v = mean(titanic_clean$Fare), col = "red", lwd = 2, lty = 2)
abline(v = median(titanic_clean$Fare), col = "orange", lwd = 2, lty = 2)
legend("topright", 
       legend = c("Mean", "Median"), 
       col = c("red", "orange"), 
       lty = 2, lwd = 2,
       cex = 0.8)

3.2 Correlation Matrix

A correlation matrix shows the strength and direction of relationships between variables.

• Values range from -1 to +1
• +1 = perfect positive correlation (both variables increase together)
• -1 = perfect negative correlation (one increases, the other decreases)
• 0 = no linear relationship
• The diagonal is always 1 (a variable perfectly correlates with itself)

cor_matrix <- cor(titanic_clean)

cor_matrix %>%
  round(4) %>%
  kable(caption = "Correlation Matrix") %>%
  kableExtra::kable_styling(bootstrap_options = c("striped", "hover", "condensed"))

corrplot(cor_matrix, 
         method = "color", 
         type = "upper",
         addCoef.col = "black",
         tl.col = "black",
         tl.srt = 45,
         number.cex = 1,
         title = "Correlation Matrix - Titanic Dataset",
         mar = c(0,0,2,0),
         col = colorRampPalette(c("#6D9EC1", "white", "#E46726"))(200))

cor_matrix_no_diag <- cor_matrix
diag(cor_matrix_no_diag) <- NA

max_cor <- which(abs(cor_matrix_no_diag) == max(abs(cor_matrix_no_diag), na.rm = TRUE), arr.ind = TRUE)
max_cor_value <- cor_matrix[max_cor[1,1], max_cor[1,2]]

cat("Strongest Correlation:\n",
    rownames(cor_matrix)[max_cor[1,1]], "vs", 
    colnames(cor_matrix)[max_cor[1,2]], "=", 
    round(max_cor_value, 4), "\n\n")

## Strongest Correlation:
##  Parch vs SibSp = 0.3838

Based on the correlation matrix, we can observe several interesting patterns:

1. Parch vs SibSp (positive correlation): This makes sense because both variables represent family size. Passengers traveling with siblings/spouses often also travel with parents/children.

2. Age vs SibSp (negative correlation): Older passengers tend to travel with fewer siblings/spouses. This is logical - as people age, they’re less likely to travel with large family groups.

3. Age vs Parch (negative correlation): Similar to above, older passengers have fewer parents/children aboard. This is expected since older passengers likely have independent adult children who aren’t traveling with them.

4. Fare vs other variables : Ticket prices don’t strongly correlate with demographic variables. This suggests that fare was more related to cabin class than to passenger characteristics like age or family size.

3.3 Variance-Covariance Matrix

This matrix contains two types of information:

• Diagonal elements (variance): Show how spread out each variable’s data is
• Off-diagonal elements (covariance): Show how two variables change together

If correlation tells us the direction and strength of a relationship, covariance tells us that plus the scale of the relationship.

Formula:

• Variance:

  σ² = Σ(xᵢ - x̄)² / (n-1)

• Covariance:

  Cov(X,Y) = Σ(xᵢ - x̄)(yᵢ - ȳ) / (n-1)

Key difference from correlation: Covariance is not standardized, so its values depend on the units of measurement. Correlation is standardized (-1 to +1).

cov_matrix <- cov(titanic_clean)

cov_matrix %>%
  round(4) %>%
  kable(caption = "Variance-Covariance Matrix") %>%
  kableExtra::kable_styling(bootstrap_options = c("striped", "hover", "condensed"))

variances <- diag(cov_matrix)

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

variance_df %>%
  kable(caption = "Variance and Standard Deviation of Each Variable", digits = 4) %>%
  kableExtra::kable_styling(bootstrap_options = c("striped", "hover"))

corrplot(cov_matrix, 
         method = "color", 
         is.corr = FALSE,
         addCoef.col = "black",
         tl.col = "black",
         tl.srt = 45,
         number.cex = 0.8,
         title = "Variance-Covariance Matrix",
         mar = c(0,0,2,0),
         col = colorRampPalette(c("#6D9EC1", "white", "#E46726"))(200))

Note: Unlike the correlation plot, colors here represent the magnitude of covariance (not standardized), so variables with larger scales will have more intense colors.

Key findings from the variance-covariance matrix:

1. Fare has the highest variance (2800,41): This indicates huge variability in ticket prices. Some passengers paid very little while others paid a lot, reflecting different cabin classes (1st, 2nd, 3rd class). The standard deviation is about £52,9, which is quite large.

2. Age has moderate variance (≈211): Ages range widely, from young children to elderly passengers. Standard deviation of ≈14.5 years means there’s good age diversity.

3. SibSp and Parch have low variance: Most passengers traveled alone or with few family members. Values close to 0 are most common, with occasional larger families.

4. Covariances confirm correlation patterns:

   • Positive Cov(SibSp, Parch) = family members travel together
   • Negative Cov(Age, SibSp) = older passengers have fewer siblings/spouses aboard

3.4 Eigenvalues and Eigenvectors

Imagine we have a dataset with multiple variables plotted in multi-dimensional space. Eigenvalues and eigenvectors help us find the “principal directions” in this space - the directions where data varies the most.

• Eigenvector: A direction/axis in the data
• Eigenvalue: How much variance exists along that direction

In Principal Component Analysis (PCA), we use eigenvalues and eigenvectors to:

1. Reduce dimensions: Transform 4 variables into 2-3 “principal components”
2. Remove noise: Keep components with high eigenvalues, discard those with low eigenvalues
3. Visualize data: Plot high-dimensional data in 2D or 3D
4. Avoid multicollinearity: Create independent components for regression analysis

A common rule is to keep only components with eigenvalue > 1. These components explain more variance than a single original variable.

eigen_cor <- eigen(cor_matrix)

prop_var_cor <- eigen_cor$values / sum(eigen_cor$values) * 100

eigen_summary_cor <- data.frame(
  PC = paste0("PC", 1:length(eigen_cor$values)),
  Eigenvalue = eigen_cor$values,
  Proportion = prop_var_cor,
  Cumulative = cumsum(prop_var_cor)
)

eigen_summary_cor %>%
  kable(caption = "Eigenvalues dan Proporsi Varians (Correlation Matrix)", 
        digits = 4,
        col.names = c("Principal Component", "Eigenvalue", "Proportion (%)", "Cumulative (%)")) %>%
  kableExtra::kable_styling(bootstrap_options = c("striped", "hover"))

eigen_vectors_cor <- eigen_cor$vectors
rownames(eigen_vectors_cor) <- colnames(cor_matrix)
colnames(eigen_vectors_cor) <- paste0("PC", 1:ncol(eigen_vectors_cor))

eigen_vectors_cor %>%
  round(4) %>%
  kable(caption = "Eigenvectors (Correlation Matrix)") %>%
  kableExtra::kable_styling(bootstrap_options = c("striped", "hover", "condensed"))

eigen_cov <- eigen(cov_matrix)

prop_var_cov <- eigen_cov$values / sum(eigen_cov$values) * 100

eigen_summary_cov <- data.frame(
  PC = paste0("PC", 1:length(eigen_cov$values)),
  Eigenvalue = eigen_cov$values,
  Proportion = prop_var_cov,
  Cumulative = cumsum(prop_var_cov)
)

eigen_summary_cov %>%
  kable(caption = "Eigenvalues dan Proporsi Varians (Covariance Matrix)", 
        digits = 4,
        col.names = c("Principal Component", "Eigenvalue", "Proportion (%)", "Cumulative (%)")) %>%
  kableExtra::kable_styling(bootstrap_options = c("striped", "hover"))

eigen_vectors_cov <- eigen_cov$vectors
rownames(eigen_vectors_cov) <- colnames(cov_matrix)
colnames(eigen_vectors_cov) <- paste0("PC", 1:ncol(eigen_vectors_cov))

eigen_vectors_cov %>%
  round(4) %>%
  kable(caption = "Eigenvectors (Covariance Matrix)") %>%
  kableExtra::kable_styling(bootstrap_options = c("striped", "hover", "condensed"))

par(mfrow = c(1, 2))

plot(1:length(eigen_cor$values), eigen_cor$values, 
     type = "b", 
     pch = 19,
     col = "blue",
     main = "Scree Plot (Correlation Matrix)",
     xlab = "Principal Component",
     ylab = "Eigenvalue",
     cex.main = 1.3,
     lwd = 2)
abline(h = 1, col = "red", lty = 2, lwd = 2)
text(3, 1.1, "Kaiser Criterion (Eigenvalue = 1)", col = "red", cex = 0.8)
grid()

plot(1:length(eigen_cov$values), eigen_cov$values, 
     type = "b", 
     pch = 19,
     col = "darkgreen",
     main = "Scree Plot (Covariance Matrix)",
     xlab = "Principal Component",
     ylab = "Eigenvalue",
     cex.main = 1.3,
     lwd = 2)
grid()