Module 05: Dimensionality Reduction - Participation Exercises

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Module 05: Dimensionality Reduction - Participation Exercises#

Exercise 5.1: Prediction - Variance Explained#

Type: 🔮 Prediction (3 min)

You have a dataset with 10 features describing chemical compounds. You apply PCA.

Predict: How much variance do you think PC1 will explain?

  • < 20% (features are independent)

  • 20-40% (some correlation)

  • 40-60% (moderate correlation)

  • > 60% (highly correlated features)

Consider: What does your answer imply about the “true” dimensionality of the data?

Your prediction and reasoning:

Exercise 5.2: Mini-Exercise - Interpret PCA Loadings#

Type: 🔧 Mini-Exercise (7 min)

Analyze PCA loadings to understand what each component represents.

import numpy as np
import pandas as pd
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler

# Polymer property data (synthetic)
np.random.seed(42)
n = 100

# Create correlated features
strength = np.random.normal(50, 10, n)
hardness = strength * 0.8 + np.random.normal(0, 5, n)  # Correlated with strength
flexibility = 100 - strength + np.random.normal(0, 8, n)  # Anti-correlated
density = np.random.normal(1.2, 0.2, n)  # Independent
cost = np.random.normal(10, 3, n)  # Independent

df = pd.DataFrame({
    'strength': strength,
    'hardness': hardness,
    'flexibility': flexibility,
    'density': density,
    'cost': cost
})

# Apply PCA
X_scaled = StandardScaler().fit_transform(df)
pca = PCA()
pca.fit(X_scaled)

# Look at loadings
loadings = pd.DataFrame(
    pca.components_.T,
    columns=[f'PC{i+1}' for i in range(5)],
    index=df.columns
)
print("PCA Loadings:")
print(loadings.round(3))

print("\nVariance explained:", pca.explained_variance_ratio_.round(3))

# TASK: Interpret what PC1 and PC2 represent based on the loadings
# What physical meaning can you assign to each component?

PCA Loadings:
               PC1    PC2    PC3    PC4    PC5
strength     0.600  0.071  0.125  0.206  0.759
hardness     0.567  0.155 -0.009  0.536 -0.607
flexibility -0.517 -0.143 -0.027  0.819  0.204
density     -0.203  0.632  0.748  0.014 -0.026
cost        -0.103  0.742 -0.652  0.014  0.116

Variance explained: [0.483 0.232 0.156 0.097 0.032]

Your interpretation:

  • PC1 represents:

  • PC2 represents:

Exercise 5.3: Discussion - PCA vs t-SNE#

Type: 💬 Discussion (5 min)

You need to visualize a high-dimensional dataset. When would you choose:

  1. PCA over t-SNE?

  2. t-SNE over PCA?

  3. Both (for different purposes)?

Consider: interpretability, reproducibility, global vs local structure, computational cost.

Discussion notes:

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