AdvancedQuantitative MethodsPython

Run this module

cd "Quantitative Methods - Principal Component Analysis"
python "pca.py"

Principal Component Analysis (PCA)¶

PCA finds the orthogonal directions that explain the most variance in a dataset. In finance it powers yield-curve decomposition (level/slope/curvature), statistical factor extraction, dimensionality reduction, and covariance de-noising.

Functions¶

Function	Description
`standardize(data)`	Z-score columns to mean 0, unit variance (for correlation-matrix PCA)
`pca(data, n_components=None)`	Eigendecomposition → components, eigenvalues, variance ratios, scores
`reconstruct(scores, components, mean)`	Rebuild data from a (truncated) component set
`cumulative_variance(ratios)`	Running cumulative variance explained (scree / elbow analysis)

Key Concepts¶

PCA = eigendecomposition of the covariance matrix. Eigenvectors are the component directions (loadings); eigenvalues are the variance each explains.
Yield curves: the first three PCs are reliably interpretable as level (parallel shift), slope (steepening/flattening), and curvature (bowing) — together they typically explain >99% of variation.
Variance ratio: eigenvalue / total variance tells you how much each component matters; use the cumulative sum to pick how many to keep.
Low-rank reconstruction: keep the top-k components and discard the rest to de-noise a covariance matrix or compress correlated returns.

Example¶

import numpy as np
from pca import pca, cumulative_variance

returns = np.random.default_rng(0).normal(0, 0.01, (500, 8))
result = pca(returns, n_components=3)

print(result["explained_variance_ratio"])     # variance per PC
print(cumulative_variance(result["explained_variance_ratio"]))
print(result["scores"].shape)                  # (500, 3) factor time series

Practical Notes¶

Use np.linalg.eigh (symmetric solver), not eig — it is faster and returns real, orthonormal eigenvectors.
Standardise first (standardize) when variables have different units, so you analyse the correlation matrix rather than the covariance matrix.
Eigenvector signs are arbitrary; interpret loadings by their relative signs and magnitudes, not absolute sign.

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