Understanding PCA Foundations

Introduction

Embark on an exciting journey through the world of Principal Component Analysis (PCA). In this lesson, we will explore the indispensable roles of eigenvalues and eigenvectors in understanding the PCA framework and dive into the computation of these mathematical constructs using R . Our adventure will cover the essential role of the covariance matrix and how to compute it. Ready? Set? Let's start!

Collecting Data

At the onset, we start with a dataset housing different physical measures: weight_lbs (in lbs), height_inches (in inches), and height_cm (in cm). We capture these in R using vectors and combine them into a data frame for easy manipulation: R # Given data weight_lbs <- c(150, 160, 155, 165, 170, 160, 158, 175, 180, 170) height_inches <- c(68, 72, 66, 69, 71, 65, 67, 70, 73, 68) height_cm <- c(172.72, 182.88, 167.64, 175.26, 180.34, 165.1, 170.18, 177.8, 185.42, 172.72) # Create a data frame df <- data.frame(weight_lbs = weight_lbs, height_inches = height_inches, height_cm = height_cm) # Given data weight_lbs <- c(150, 160, 155, 165, 170, 160, 158, 175, 180, 170) height_inches <- c(68, 72, 66, 69, 71, 65, 67, 70, 73, 68) height_cm <- c(172.72, 182.88, 167.64, 175.26, 180.34, 165.1, 170.18, 177.8, 185.42, 172.72) # Create a data frame df <- data.frame(weight_lbs = weight_lbs, height_inches = height_inches, height_cm = height_cm) Here, the data frame df represents our collected dataset.

Introduction to Standardization

X_{standardized} = \frac{X - \mu}{\sigma}

Plotting Standardized Data

Let's standardize just the two height columns in our dataset using R's scale() function and visualize the standardized data using ggplot2 : R library(ggplot2) # Select the height columns and standardize X <- df[, c("height_inches", "height_cm")] X_standard <- scale(X) # Convert to data frame for plotting X_standard_df <- as.data.frame(X_standard) colnames(X_standard_df) <- c("height_inches", "height_cm") # Plot standardized data ggplot(X_standard_df, aes(x = height_inches, y = height_cm)) + geom_point(color = "blue") + ggtitle("Standardized Data") + xlab("height_inches") + ylab("height_cm") + theme_bw() + theme(panel.grid = element_line()) library(ggplot2) # Select the height columns and standardize X <- df[, c("height_inches", "height_cm")] X_standard <- scale(X) # Convert to data frame for plotting X_standard_df <- as.data.frame(X_standard) colnames(X_standard_df) <- c("height_inches", "height_cm") # Plot standardized data ggplot(X_standard_df, aes(x = height_inches, y = height_cm)) + geom_point(color = "blue") + ggtitle("Standardized Data") + xlab("height_inches") + ylab("height_cm") + theme_bw() + theme(panel.grid = element_line()) Output: After standardization, our data is now centered and scaled, making the variables more comparable.

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# Given data
weight_lbs <- c(150, 160, 155, 165, 170, 160, 158, 175, 180, 170)
height_inches <- c(68, 72, 66, 69, 71, 65, 67, 70, 73, 68)
height_cm <- c(172.72, 182.88, 167.64, 175.26, 180.34, 165.1, 170.18, 177.8, 185.42, 172.72)

# Create a data frame
df <- data.frame(
  weight_lbs = weight_lbs,
  height_inches = height_inches,
  height_cm = height_cm
)

library(ggplot2)

# Select the height columns and standardize
X <- df[, c("height_inches", "height_cm")]
X_standard <- scale(X)

# Convert to data frame for plotting
X_standard_df <- as.data.frame(X_standard)
colnames(X_standard_df) <- c("height_inches", "height_cm")

# Plot standardized data
ggplot(X_standard_df, aes(x = height_inches, y = height_cm)) +
  geom_point(color = "blue") +
  ggtitle("Standardized Data") +
  xlab("height_inches") +
  ylab("height_cm") +
  theme_bw() +
  theme(panel.grid = element_line())

# Compute the covariance matrix
cov_matrix <- cov(X_standard)

cat("Covariance Matrix:\n")
print(cov_matrix)

# Compute the covariance matrix
cov_matrix <- cov(X_standard)

cat("Covariance Matrix:\n")
print(cov_matrix)

              height_inches height_cm
height_inches      1.111111  1.111111
height_cm          1.111111  1.111111

              height_inches height_cm
height_inches      1.111111  1.111111
height_cm          1.111111  1.111111

# Eigendecomposition
eig <- eigen(cov_matrix)
eigenvalues <- eig$values
eigenvectors <- eig$vectors

cat("\nEigenvalues:\n")
print(eigenvalues)

cat("\nEigenvectors:\n")
print(eigenvectors)

# Eigendecomposition
eig <- eigen(cov_matrix)
eigenvalues <- eig$values
eigenvectors <- eig$vectors

cat("\nEigenvalues:\n")
print(eigenvalues)

cat("\nEigenvectors:\n")
print(eigenvectors)

Eigenvalues:
[1] 2.222222e+00 1.110223e-16

Eigenvectors:
           [,1]       [,2]
[1,] 0.7071068 -0.7071068
[2,] 0.7071068  0.7071068

Eigenvalues:
[1] 2.222222e+00 1.110223e-16

Eigenvectors:
           [,1]       [,2]
[1,] 0.7071068 -0.7071068
[2,] 0.7071068  0.7071068

library(ggplot2)

# Prepare data for plotting
origin <- data.frame(x = 0, y = 0)
eigvec_df <- data.frame(
  xend = eigenvectors[1, ],
  yend = eigenvectors[2, ],
  label = c("Eigenvector 1", "Eigenvector 2"),
  color = c("red", "green")
)

# Plot standardized data and eigenvectors
ggplot(X_standard_df, aes(x = height_inches, y = height_cm)) +
  geom_point(color = "blue") +
  geom_segment(
    data = eigvec_df,
    aes(x = 0, y = 0, xend = xend, yend = yend, color = label),
    arrow = arrow(length = unit(0.3, "cm")), linewidth = 1.2
  ) +
  scale_color_manual(values = c("red", "green")) +
  ggtitle("Eigenvectors of Covariance Matrix") +
  xlab("height_inches") +
  ylab("height_cm") +
  theme_bw() +
  theme(panel.grid = element_line())

library(ggplot2)

# Prepare data for plotting
origin <- data.frame(x = 0, y = 0)
eigvec_df <- data.frame(
  xend = eigenvectors[1, ],
  yend = eigenvectors[2, ],
  label = c("Eigenvector 1", "Eigenvector 2"),
  color = c("red", "green")
)

# Plot standardized data and eigenvectors
ggplot(X_standard_df, aes(x = height_inches, y = height_cm)) +
  geom_point(color = "blue") +
  geom_segment(
    data = eigvec_df,
    aes(x = 0, y = 0, xend = xend, yend = yend, color = label),
    arrow = arrow(length = unit(0.3, "cm")), linewidth = 1.2
  ) +
  scale_color_manual(values = c("red", "green")) +
  ggtitle("Eigenvectors of Covariance Matrix") +
  xlab("height_inches") +
  ylab("height_cm") +
  theme_bw() +
  theme(panel.grid = element_line())

# Compute the covariance matrix
cov_matrix <- cov(X_standard)

cat("Covariance Matrix:\n")
print(cov_matrix)

              height_inches height_cm
height_inches      1.111111  1.111111
height_cm          1.111111  1.111111

# Eigendecomposition
eig <- eigen(cov_matrix)
eigenvalues <- eig$values
eigenvectors <- eig$vectors

cat("\nEigenvalues:\n")
print(eigenvalues)

cat("\nEigenvectors:\n")
print(eigenvectors)

Eigenvalues:
[1] 2.222222e+00 1.110223e-16

Eigenvectors:
           [,1]       [,2]
[1,] 0.7071068 -0.7071068
[2,] 0.7071068  0.7071068

library(ggplot2)

# Prepare data for plotting
origin <- data.frame(x = 0, y = 0)
eigvec_df <- data.frame(
  xend = eigenvectors[1, ],
  yend = eigenvectors[2, ],
  label = c("Eigenvector 1", "Eigenvector 2"),
  color = c("red", "green")
)

# Plot standardized data and eigenvectors
ggplot(X_standard_df, aes(x = height_inches, y = height_cm)) +
  geom_point(color = "blue") +
  geom_segment(
    data = eigvec_df,
    aes(x = 0, y = 0, xend = xend, yend = yend, color = label),
    arrow = arrow(length = unit(0.3, "cm")), linewidth = 1.2
  ) +
  scale_color_manual(values = c("red", "green")) +
  ggtitle("Eigenvectors of Covariance Matrix") +
  xlab("height_inches") +
  ylab("height_cm") +
  theme_bw() +
  theme(panel.grid = element_line())

Introduction

Collecting Data

Introduction to Standardization

Plotting Standardized Data

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Introduction to Covariance Matrix

Building Covariance Matrix

Introduction to Eigenvalues and Eigenvectors

Deciphering Covariance Matrix with Eigendecomposition

Interpretation of Eigenvectors and Eigenvalues with an Example

Connecting Eigenvectors and Eigenvalues to PCA

Lesson Summary & Next Steps