PCA

Principal-components transformer

from _api_doc_utils import *

1 Where it fits

Group: Transforms

PCA learns an orthogonal low-rank basis from a design matrix. It centers the training data, computes principal directions, and maps observations to component scores:

\[ Z = (X - \bar X) V_k. \]

With whiten=True, scores are rescaled by the singular values.

2 Python API

Constructor: cm.PCA

Use PCA(n_components, whiten=False). fit(x) stores the basis, transform(x) maps new data to scores, fit_transform(x) combines both, and inverse_transform(scores) reconstructs approximate original features. summary() exposes components, means, explained variance, ratios, and singular values.

print(inspect.signature(cm.PCA))

(n_components, whiten=False)

cls = cm.PCA
display(HTML(html_table(["Public method"], public_methods(cls))))

Public method
fit(self, /, x)
fit_transform(self, /, x)
inverse_transform(self, /, scores)
summary(self, /)
transform(self, /, x)

3 Minimal example

rng=np.random.default_rng(22)
x=rng.normal(size=(150,5)) @ np.array([[1,.2,.1,0,0],[0,.8,.3,.1,0],[0,0,.5,.2,.1],[0,0,0,.3,.2],[0,0,0,0,.1]])
model=cm.PCA(n_components=2); scores=model.fit_transform(x)
print(scores.shape)
print(model.summary()["explained_variance_ratio"])
print(model.inverse_transform(scores[:2]))

(150, 2)
[0.62644017 0.37355983]
[[-1.40550383 -1.36130695 -1.01359223 -0.47915055 -0.19206572]
 [-1.57694267  0.65071588  0.38647668  0.29693871  0.11740384]]

4 summary() contract

The table below is generated by fitting the live class in this repository and then inspecting summary(). Shapes are shown because most values are plain NumPy arrays or scalars.

rng=np.random.default_rng(122); x=rng.normal(size=(80,5)); model=cm.PCA(n_components=2); model.fit(x)
summary = model.summary()
display(HTML(html_table(["summary() key", "shape"], summary_shape_rows(summary))))

summary() key	shape
n_components	()
n_features	()
whiten	()
components	(2, 5)
mean	(5,)
explained_variance	(2,)
explained_variance_ratio	(2,)
singular_values	(2,)