Skill: wu-uk/lake-warming-attribution-pca-decomposition (version 0.1.0)

--- name: pca-decomposition description: Reduce dimensionality of multivariate data using PCA with varimax rotation. Use when you have many correlated variables and need to identify underlying factors or reduce collinearity. license: MIT --- # 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 ```python 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: ```python # 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: | Loading | Interpretation | |---------|----------------| | > 0.7 | Strong association | | 0.4 - 0.7 | Moderate association | | < 0.4 | Weak association | ## Example: Economic Indicators ```python import pandas as pd from sklearn.preprocessing import StandardScaler from factor_analyzer import FactorAnalyzer # Variables: gdp, unemployment, inflation, interest_rate, exports, imports df = pd.read_csv('economic_data.csv') variables = ['gdp', 'unemployment', 'inflation', 'interest_rate', 'exports', 'imports'] X = df[variables].values scaler = StandardScaler() X_scaled = scaler.fit_transform(X) fa = FactorAnalyzer(n_factors=3, rotation='varimax') fa.fit(X_scaled) # View loadings loadings_df = pd.DataFrame( fa.loadings_, index=variables, columns=['RC1', 'RC2', 'RC3'] ) print(loadings_df.round(2)) ``` ## Choosing Number of Factors ### Option 1: Kaiser Criterion ```python # Check eigenvalues eigenvalues, _ = fa.get_eigenvalues() # Keep factors with eigenvalue > 1 n_factors = sum(eigenvalues > 1) ``` ### Option 2: Domain Knowledge If you know how many categories your variables should group into, specify directly: ```python # Example: health data with 3 expected categories (lifestyle, genetics, environment) fa = FactorAnalyzer(n_factors=3, rotation='varimax') ``` ## Common Issues | Issue | Cause | Solution | |-------|-------|----------| | Loadings all similar | Too few factors | Increase n_factors | | Negative loadings | Inverse relationship | Normal, interpret direction | | Low variance explained | Data not suitable for PCA | Check correlations first | ## Best Practices - Always standardize data before PCA - Use varimax rotation for interpretability - Check factor loadings to name components - Use Kaiser criterion or domain knowledge for n_factors - For attribution analysis, run ONE global PCA on all variables


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