Show students the relationship between Eigendecomp of Cor/Cov and the % variance explained for PCs

R_EigenPCA_Plots.R

library(tidyverse)
library(MASS)
library(patchwork)
cbPalette <- c("#999999", "#E69F00", "#56B4E9", "#009E73", "#F0E442", "#0072B2", "#D55E00", "#CC79A7")

# generate data with given cor matrix
a <- 0.9
s1 <- matrix(c(1,a,
               a,1), ncol = 2)

# data
df <- data.frame(mvrnorm(n = 1000, mu = c(0,0), Sigma = s1))

# find eigenvalues
pca1 <- eigen(cor(df))$values

# print matrix, eigenvals, and % var
print("Correlation Matrix: ")
print(s1)
print("Eigenvalues: ")
print(pca1)
print("% Variance Explained: ")
print(pca1/sum(pca1))

#---eigen----

# unit square
df <- data.frame(x = c(0,0,1,1),
                 y = c(0,1,0,1))

# plot unit squre
p1 <- ggplot(df, aes(x = x, y = y)) + 
  coord_fixed() + 
  geom_rect(data=df,
            mapping=aes(xmin=0, xmax=1,
                        ymin=0, ymax=1),
            color= cbPalette[2],
            fill = cbPalette[2],
            alpha=0.05) + 
  geom_point(size = 3) + 
  theme_minimal() + 
  theme(panel.grid.minor = element_blank()) + 
  lims(x = c(-0.10,3), y = c(-0.10,3)) + 
  labs(x = "", y = "",
       title = "Unit Square") +
  annotate(geom = "text", x = -0.10, y = -0.10, label = "(0,0)") +
  annotate(geom = "text", x = 1.10, y = 1.10, label = "(1,1)") + 
  annotate(geom = "text", x = -0.10, y = 1.10, label = "(0,1)") + 
  annotate(geom = "text", x = 1.10, y = -0.10, label = "(1,0)") 



# multiplied------------
# sanity check of multiplications
multis <- list(s1 %*% c(0,0), s1 %*% c(0,1), s1 %*% c(1,1),s1 %*% c(1,0))
print("Four Corners Transformed: ")
print(multis)

# new paralellogram df
df <- data.frame(x = c(0,a,(1 + a),1),
                 y = c(0,1,(1 + a),a))

# plot paralellogram
p2 <- ggplot(df, aes(x = x, y = y)) + 
  geom_rect(data=df,
            mapping=aes(xmin=0, xmax=1,
                        ymin=0, ymax=1),
            color= cbPalette[1],
            fill = cbPalette[1],
            alpha=0.05,
            linetype = "dashed") + 
  coord_fixed() + 
  geom_polygon(
    color= cbPalette[2],
    fill = cbPalette[2],
    alpha=0.2) + 
  geom_point(size = 3) + 
  theme_minimal() + 
  theme(panel.grid.minor = element_blank()) + 
  lims(x = c(-0.10,3), y = c(-0.10,3)) + 
  labs(x = "", y = "",
       title = paste0("Transformed Paralellogram ", "r = ",a )) +
  annotate(geom = "text", x = -0.10, y = -0.10, label = "(0,0)") +
  annotate(geom = "text", x = 1 + a + 0.1, y = 1 + a + 0.1, label = paste0("(", 1 + a, ",", 1 + a, ")")) +
  annotate(geom = "text", x = a - 0.1, y = 1.10, label = paste0("(", a, ",", 1 ,")")) +
  annotate(geom = "text", x = 1.10, y = a- 0.1, label = paste0("(", 1 + a, ",", a, ")"))

# plot both together
p1 + p2