PCA Decomposition Guide

Overview

Principal Component Analysis (PCA) reduces many correlated variables into fewer uncorrelated components. Varimax rotation makes components more interpretable by maximizing variance.

When to Use PCA

• Many correlated predictor variables
• Need to identify underlying factor groups
• Reduce multicollinearity before regression
• Exploratory data analysis

Basic PCA with Varimax Rotation

from sklearn.preprocessing import StandardScaler
from factor_analyzer import FactorAnalyzer

# Standardize data first
scaler = StandardScaler()
X_scaled = scaler.fit_transform(X)

# PCA with varimax rotation
fa = FactorAnalyzer(n_factors=4, rotation='varimax')
fa.fit(X_scaled)

# Get factor loadings
loadings = fa.loadings_

# Get component scores for each observation
scores = fa.transform(X_scaled)

Workflow for Attribution Analysis

When using PCA for contribution analysis with predefined categories:

1. Combine ALL variables first, then do PCA together:

# Include all variables from all categories in one matrix
all_vars = ['AirTemp', 'NetRadiation', 'Precip', 'Inflow', 'Outflow',
            'WindSpeed', 'DevelopedArea', 'AgricultureArea']
X = df[all_vars].values

scaler = StandardScaler()
X_scaled = scaler.fit_transform(X)

# PCA on ALL variables together
fa = FactorAnalyzer(n_factors=4, rotation='varimax')
fa.fit(X_scaled)
scores = fa.transform(X_scaled)

2. Interpret loadings to map factors to categories (optional for understanding)

3. Use factor scores directly for R² decomposition

Important: Do NOT run separate PCA for each category. Run one global PCA on all variables, then use the resulting factor scores for contribution analysis.

Interpreting Factor Loadings

Loadings show correlation between original variables and components: