PCA and Model Performance Trade-offs

In real-world ML projects, most of the effort is not spent on model building, but on data preprocessing, feature engineering, and dimensionality reduction. A well-prepared dataset often matters more than the choice of algorithm.
One of the most powerful techniques for reducing complexity while retaining information is Principal Component Analysis (PCA).
In this article, you’ll learn:
Why dimensionality reduction is essential
The intuition behind PCA
How PCA works mathematically
A step-by-step implementation of PCA in R
How PCA impacts model performance using a real example

Table of Contents
Lifting the Curse Using Principal Component Analysis
Curse of Dimensionality in Layman’s Terms
Insights from Shlens’ PCA Paper
PCA Conceptual Background
Implementing PCA in R
Loading the Iris Dataset
Covariance Matrix
Eigenvalues and Eigenvectors
PCA Using princomp()
Interpreting Components
Visualizing PCA Results
PCA and Model Performance Trade-offs
Summary and Limitations of PCA

Lifting the Curse Using Principal Component Analysis
A common misconception in machine learning is:
“More features and more data always improve model accuracy.”
In reality, adding more features—especially with limited data—often hurts performance. This phenomenon is known as the curse of dimensionality.
When datasets contain many features but relatively few observations:
Models overfit
Distance-based methods degrade
Computational cost increases
Generalization suffers
PCA helps mitigate this by compressing information into fewer, more meaningful dimensions.

Curse of Dimensionality in Layman’s Terms
Simply put, the curse of dimensionality means:
As the number of features increases, model accuracy can decrease.
Why?
Feature space grows exponentially
Data becomes sparse
Noise dominates signal
Two Ways to Address This:
Collect more data (often impractical)
Reduce the number of features (preferred)
Reducing features without losing information is called dimensionality reduction—and PCA is one of the most widely used techniques.

Shlens’ Paper: PCA Explained Intuitively
David Shlens’ famous paper explains PCA using a simple analogy: tracking a pendulum.
Imagine a pendulum swinging in one direction:
If you know the direction, one camera is enough
If you don’t, you may need several cameras placed orthogonally
Each camera captures redundant information. PCA finds the optimal angles (principal components) that capture maximum variance with minimum views.
PCA transforms correlated variables into a smaller set of orthogonal (uncorrelated) components that retain most of the information.

PCA Conceptual Background
Suppose you have:
m observations
n features
Your dataset is an m × n matrix A.
PCA transforms A into a new matrix A′ (m × k) where k < n, such that:
New features are orthogonal
Variance is maximized
Information loss is minimized
Why Eigenvectors?
Eigenvectors are orthogonal
They define new axes of maximum variance
Eigenvalues indicate how much variance each axis explains
Why Scaling Matters
PCA is scale-sensitive. Without normalization:
Features with larger numeric ranges dominate
Results become misleading
This is why experienced AI consulting teams emphasize data normalization and governance before PCA.

Implementing PCA in R
Let’s now implement PCA step-by-step using the Iris dataset.

Step 1: Load the Dataset

Taking the numeric part of the IRIS dataset

data_iris <- iris[, 1:4]

The dataset contains:
150 observations
4 numerical features

Step 2: Covariance Matrix

Calculating covariance matrix

Cov_data <- cov(data_iris)

The covariance matrix captures how features vary together.

Step 3: Eigenvalues and Eigenvectors

Eigen decomposition

Eigen_data <- eigen(Cov_data)

Eigenvalues → amount of variance
Eigenvectors → direction of variance

Step 4: PCA Using Built-in Function

PCA using R's built-in function

PCA_data <- princomp(data_iris, cor = FALSE)

Step 5: Compare Variances
Eigen_data$values
PCA_data$sdev^2

The results are nearly identical, validating the PCA computation.

Step 6: Compare Loadings (Eigenvectors)
PCA_data$loadings[, 1:4]
Eigen_data$vectors

The loadings match exactly, confirming consistency.

Understanding PCA Results
summary(PCA_data)

Importance of Components
ComponentVariance ExplainedCumulative
PC1
~92.5%
92.5%
PC2
~5.3%
97.7%
PC3
~1.7%
99.4%
PC4
~0.5%
100%
Key Insight:
One component captures most of the information
Two components explain nearly 98% of the variance

Visualizing PCA
Biplot
biplot(PCA_data)

Petal Length & Petal Width dominate PC1
Sepal features contribute more to PC2
Scree Plot
screeplot(PCA_data, type = "lines")

The “elbow” appears at PC2, suggesting two components are sufficient.

PCA and Model Performance Trade-Off
Let’s compare:
Naive Bayes using all features
Naive Bayes using only the first principal component

First principal component

model2 <- PCA_data$loadings[,1]
model2_scores <- as.matrix(data_iris) %*% model2

library(e1071)

Full feature model

mod1 <- naiveBayes(iris[,1:4], iris[,5])

PCA-based model

mod2 <- naiveBayes(model2_scores, iris[,5])

Accuracy Comparison
PCA model misclassifies only 3 more instances
Uses 75% fewer features
This demonstrates PCA’s efficiency–accuracy trade-off.

Summary
Why PCA Works
Reduces dimensionality
Improves computational efficiency
Removes multicollinearity
Retains most variance
Limitations
Sensitive to scaling
Components lack direct business interpretation
Assumes linear relationships
Relies on variance (mean-based)
When to Use PCA
High-dimensional datasets
Image compression
Genomics
Feature engineering before modeling

Final Thoughts
PCA is a foundational technique every data scientist should master. While it simplifies data, it should be applied thoughtfully, with proper scaling, domain understanding, and validation.
When implemented correctly, PCA delivers simpler models, faster training, and strong performance—making it a valuable tool in both research and production analytics.
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