“Give me six hours to chop down a tree and I will spend the first four sharpening the axe.”
— Abraham Lincoln
This quote resonates strongly with modern machine learning and data science. In real-world projects, the majority of time is not spent on modeling, but on data preprocessing, feature engineering, and dimensionality reduction.
One of the most powerful and widely used dimensionality reduction techniques is Principal Component Analysis (PCA). PCA helps us transform high-dimensional data into a smaller, more informative feature space—often improving model performance, interpretability, and computational efficiency.
In this article, you will learn the conceptual foundations of PCA and how to implement PCA in R using modern, industry-standard practices.
Table of Contents
Lifting the Curse with Principal Component Analysis
Curse of Dimensionality in Simple Terms
Key Insights from Shlens’ PCA Perspective
Conceptual Background of PCA
Implementing PCA in R (Modern Approach)
Loading and Preparing the Iris Dataset
Scaling and Standardization
Covariance Matrix and Eigen Decomposition
PCA with prcomp()
Understanding PCA Outputs
Variance Explained
Loadings and Scores
Scree Plot and Biplot
PCA in a Modeling Workflow (Naive Bayes Example)
Summary and Practical Takeaways
- Lifting the Curse with Principal Component Analysis
A common myth in analytics is:
“More features and more data will always improve model accuracy.”
In practice, this is often false.
When the number of features grows faster than the number of observations, models become unstable, harder to generalize, and prone to overfitting. This phenomenon is known as the curse of dimensionality.
PCA helps address this issue by reducing the dimensionality of data while preserving most of its informational content.
- Curse of Dimensionality in Simple Terms
In layman’s language, the curse of dimensionality means:
Adding more features can decrease model accuracy
Model complexity grows exponentially
Distance-based and probabilistic models degrade rapidly
There are two general ways to mitigate this:
Collect more data (often expensive or impossible)
Reduce the number of features (preferred and practical)
Dimensionality reduction techniques like PCA fall into the second category.
- Shlens’ Perspective on PCA
In his well-known paper, Jonathon Shlens describes PCA using a simple analogy: observing the motion of a pendulum.
If the pendulum moves in one direction but we don’t know that direction, we may need several cameras (features) to capture its motion. PCA helps us rotate the coordinate system so that we capture the motion with fewer, orthogonal views.
In essence, PCA:
Transforms correlated variables into uncorrelated (orthogonal) components
Orders these components by variance explained
Allows us to retain only the most informative components
- PCA: Conceptual Background
Assume a dataset with:
m observations
n features
This can be represented as an m × n matrix A.
PCA transforms A into a new matrix A′ of size m × k, where k < n.
Key ideas:
PCA relies on eigenvectors and eigenvalues
Eigenvectors define new axes (principal components)
Eigenvalues represent variance captured along those axes
Components are orthogonal and uncorrelated
Why Scaling Matters
PCA is scale-sensitive. Variables with larger units dominate variance.
Modern best practice:
Always standardize features unless units are naturally comparable
Perform PCA on the correlation matrix, not raw covariance, for most ML tasks
- Implementing PCA in R (Modern Approach)
Loading and Preparing the Iris Dataset
Load numeric features only
data_iris <- iris[, 1:4]
The Iris dataset contains:
150 observations
4 numeric features
3 species (target variable)
Scaling the Data (Industry Standard)
data_scaled <- scale(data_iris)
Covariance Matrix and Eigen Decomposition
cov_data <- cov(data_scaled)
eigen_data <- eigen(cov_data)
Eigenvalues indicate variance explained by each component.
Performing PCA with prcomp()
Why prcomp()?
prcomp() is now preferred over princomp() because it:
Uses singular value decomposition (SVD)
Is numerically more stable
Works better for high-dimensional data
pca_data <- prcomp(data_iris, scale. = TRUE)
- Understanding PCA Outputs
Variance Explained
summary(pca_data)
Example output:
PC1 explains ~92% variance
PC2 explains ~5% variance
First two components explain ~97% variance cumulatively
This means we can reduce 4 features → 2 components with minimal information loss.
Loadings (Feature Contributions)
pca_data$rotation
Loadings show how original features contribute to each principal component.
Visualizations
Scree Plot
screeplot(pca_data, type = "lines")
The “elbow” typically indicates the optimal number of components.
Biplot
biplot(pca_data, scale = 0)
The biplot reveals:
Feature directions
Component importance
Correlations between variables
- PCA in a Modeling Workflow (Naive Bayes Example)
Baseline Model (All Features)
library(e1071)
model_full <- naiveBayes(iris[, 1:4], iris[, 5])
pred_full <- predict(model_full, iris[, 1:4])
table(pred_full, iris[, 5])
Model Using First Principal Component
pc_scores <- pca_data$x[, 1, drop = FALSE]
model_pca <- naiveBayes(pc_scores, iris[, 5])
pred_pca <- predict(model_pca, pc_scores)
table(pred_pca, iris[, 5])
Result
Slight reduction in accuracy
75% reduction in feature space
Faster training and simpler model
This tradeoff is often acceptable—and desirable—in production systems.
- Summary and Practical Takeaways
PCA remains one of the most important tools in modern data science.
Strengths
Effective dimensionality reduction
Removes multicollinearity
Improves model stability and performance
Widely used in image processing, genomics, NLP, and finance
Limitations
Sensitive to scaling
Components may lack business interpretability
Captures only linear relationships
Mean and variance dependent
Best Practices (2025+)
Always scale features
Use prcomp() instead of princomp()
Combine PCA with cross-validation
Apply PCA inside modeling pipelines, not before data splitting
Final Thoughts
PCA is not just a mathematical trick—it is a practical engineering tool. When used thoughtfully, it allows you to build simpler, faster, and more robust machine learning systems without sacrificing accuracy.
Just like sharpening the axe, investing time in feature engineering and dimensionality reduction pays off exponentially.
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