Principal Component Analysis Cheat Sheet

A cheat sheet for Principal Component Analysis covering scikit-learn implementation, explained variance, choosing component counts, and reconstruction error.

2 PagesIntermediateMar 12, 2026

PCA with scikit-learn

Reduce dimensionality and inspect explained variance.

python

from sklearn.decomposition import PCAfrom sklearn.preprocessing import StandardScalerX_scaled = StandardScaler().fit_transform(X)   # PCA is scale-sensitivepca = PCA(n_components=2)X_pca = pca.fit_transform(X_scaled)print('Explained variance ratio:', pca.explained_variance_ratio_)print('Total variance captured:', pca.explained_variance_ratio_.sum())

Choosing the Number of Components

Use a scree plot or a variance target.

python

import numpy as npimport matplotlib.pyplot as pltpca_full = PCA().fit(X_scaled)cumulative = np.cumsum(pca_full.explained_variance_ratio_)plt.plot(cumulative)plt.xlabel('Number of components'); plt.ylabel('Cumulative explained variance')# Or let sklearn pick components that explain 95% of the variancepca_95 = PCA(n_components=0.95).fit(X_scaled)print(pca_95.n_components_)

Reconstruction

Project back to the original feature space.

python

X_reduced = pca.transform(X_scaled)X_reconstructed = pca.inverse_transform(X_reduced)   # lossy reconstructionreconstruction_error = ((X_scaled - X_reconstructed) ** 2).mean()

Key Concepts

Core theory behind PCA.

Principal components- Orthogonal directions of maximum variance in the data, ordered by how much variance they explain
Eigenvectors/eigenvalues- Components are eigenvectors of the covariance matrix; eigenvalues indicate variance captured along each
Explained variance ratio- Fraction of total dataset variance captured by each principal component
Dimensionality reduction- Projecting onto the top-k components reduces feature count while preserving most information
Standardization- Features must be scaled first, or high-variance features will dominate the components
Whitening- whiten=True rescales components to unit variance, useful before some downstream algorithms

Pro Tip

PCA components are linear combinations of all original features, which makes them hard to interpret directly — inspect pca.components_ (the loadings) to see which original features contribute most to each principal component.

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