particel applications

library(mvtnorm)

library(ggplot2)

library(matlib)

set.seed(123)

n <- 100 

mu <- c(1, 2)

sigma <- matrix(c(1, 0.5, 0.5, 2), nrow = 2) 

data <- rmvnorm(n, mean = mu, sigma = sigma)

pca_result <- prcomp(data)

pc_scores <- pca_result$x  

pc_loadings <- pca_result$rotation 

variance_ratio <- pca_result$sdev^2 / sum(pca_result$sdev^2)
cat("Explained Variance Ratio:", variance_ratio, "\n")

## Explained Variance Ratio: 0.7308674 0.2691326

pca_plot <- ggplot(data.frame(pc_scores), aes(x = PC1, y = PC2)) +
  geom_point() +
  xlab("Principal Component 1") +
  ylab("Principal Component 2") +
  ggtitle("PCA Scatter Plot")

pca_plot + stat_ellipse(level = 0.95, col = "blue", type = "t")