Đại số tuyến tính 7: vector riêng, hiệp phương sai và PCA

py -m venv .venv
.\.venv\Scripts\python.exe -m pip install numpy matplotlib

from pathlib import Path
import json
import numpy as np
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt
OUTPUT = Path(__file__).resolve().parent / "results"
OUTPUT.mkdir(exist_ok=True)
plt.rcParams.update({"font.size": 12, "axes.spines.top": False,
                     "axes.spines.right": False, "figure.facecolor": "#f3f7f6",
                     "axes.facecolor": "white", "axes.titleweight": "bold"})
FIGURE = OUTPUT / "linear-pca.png"

rng=np.random.default_rng(71)
base=rng.normal(size=(300,2))
X=base@np.array([[2.,1.2],[0.2,0.6]])+np.array([4.,-2.])
mean=X.mean(axis=0)
centered=X-mean
cov=centered.T@centered/(len(X)-1)
values,vectors=np.linalg.eigh(cov)
order=np.argsort(values)[::-1]
values,vectors=values[order],vectors[:,order]
assert np.allclose(cov@vectors,vectors*values)
assert np.all(values>=-1e-12)

direction=vectors[:,:1]
scores=centered@direction
reconstructed=scores@direction.T+mean
ratio=values/values.sum()
error=np.mean(np.sum((X-reconstructed)**2,axis=1))
expected_error=values[1]*(len(X)-1)/len(X)
assert np.isclose(error,expected_error)
assert np.allclose(scores@direction.T,(-scores)@(-direction).T)
print("Explained variance ratios:",ratio)

fig,axs=plt.subplots(1,2,figsize=(12,8),layout="constrained")
axs[0].scatter(X[:,0],X[:,1],s=10,alpha=0.35,color="#387a55",label="Original")
axs[0].scatter(reconstructed[:,0],reconstructed[:,1],s=10,color="#bb9457",label="Rank-one reconstruction")
axs[0].set(aspect="equal",title="PCA projection",xlabel="Feature 1",ylabel="Feature 2")
axs[0].legend()
axs[1].bar(["PC1","PC2"],ratio,color=["#387a55","#bb9457"])
axs[1].set(ylim=(0,1),ylabel="Explained variance ratio",title="Variance is not class information")
fig.savefig(FIGURE, dpi=100)
plt.close(fig)
print("Saved:", FIGURE.name)

.\.venv\Scripts\python.exe linear_pca.py