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Machine Learning Academy · Aula

Projeção de dados e reconstrução a partir de componentes

Os alunos transformarão um conjunto de dados para o espaço dos componentes principais, visualizarão a projeção 2D e reconstruirão as características originais para quantificar a perda de informação.

Projeção de dados e reconstrução a partir de componentes é uma aula grátis de Machine Learning Academy no CoddyKit. Esta é a aula 2 de 4. Você pode ler a aula completa abaixo gratuitamente — depois pratica ao vivo no navegador com um editor de código integrado e um tutor de IA 24/7. Faz parte do caminho de aprendizado de Machine Learning Academy, e seu progresso é sincronizado entre a web e o app CoddyKit. O curso de Machine Learning Academy inclui 4 aulas no total.

Partes desta aula ainda não foram traduzidas e aparecem em inglês.

Projection: From High-D to Low-D

After PCA finds the principal components, projection transforms each data point into the new component space. The projected coordinates are called scores. If you keep only 2 components from 64 original features, each 64-dimensional point becomes a 2-dimensional score. This is achieved by multiplying the centred data matrix by the matrix of eigenvectors (the loadings matrix).

The transform Method in sklearn

In scikit-learn, pca.fit(X) learns the components and pca.transform(X) projects the data. The convenience method pca.fit_transform(X) does both in one call. The result is a matrix of shape (n_samples, n_components) — each row is a point in the reduced space.

from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
from sklearn.datasets import load_digits

X, y = load_digits(return_X_y=True)  # 1797 x 64
X_scaled = StandardScaler().fit_transform(X)

pca = PCA(n_components=10)
X_reduced = pca.fit_transform(X_scaled)

print('Original shape:', X_scaled.shape)
print('Reduced shape:', X_reduced.shape)
print('Variance retained:', pca.explained_variance_ratio_.sum().round(4))

Visualising the 2D Projection

Projecting to 2 components gives a scatter plot where class separation is often visible even though labels were never used during PCA. This is an important exploratory tool: if classes are well-separated in 2D PCA space, a simple linear classifier may perform well in the full-dimensional space.

import matplotlib.pyplot as plt
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
from sklearn.datasets import load_digits

X, y = load_digits(return_X_y=True)
X_scaled = StandardScaler().fit_transform(X)

pca2 = PCA(n_components=2)
X_2d = pca2.fit_transform(X_scaled)

plt.figure(figsize=(8, 6))
for digit in range(10):
    mask = y == digit
    plt.scatter(X_2d[mask, 0], X_2d[mask, 1], label=str(digit), s=10, alpha=0.6)
plt.legend(title='Digit', bbox_to_anchor=(1, 1))
plt.title('MNIST digits in 2D PCA space')
plt.tight_layout()
plt.show()

Reconstruction: Going Back to Original Space

Reconstruction reverses the projection: multiply the reduced scores by the transpose of the loadings matrix and add back the mean. The result is an approximation of the original data in the original feature space. Perfect reconstruction is only possible if you kept all components; retaining fewer introduces reconstruction error.

from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
from sklearn.datasets import load_digits
import numpy as np

X, _ = load_digits(return_X_y=True)
X_scaled = StandardScaler().fit_transform(X)

pca = PCA(n_components=20)
X_reduced = pca.fit_transform(X_scaled)

# Reconstruct back to 64 dimensions
X_reconstructed = pca.inverse_transform(X_reduced)
print('Reconstruction shape:', X_reconstructed.shape)

# Mean squared reconstruction error
mse = np.mean((X_scaled - X_reconstructed) ** 2)
print(f'MSE: {mse:.4f}')

Visualising Reconstruction Quality

For image data, you can plot original and reconstructed images side by side. With more components retained, the reconstruction looks sharper. With very few components, digits become blurry blobs. This visual comparison is a powerful communication tool for showing stakeholders the trade-off between compression and information loss.

import matplotlib.pyplot as plt
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
from sklearn.datasets import load_digits

X, _ = load_digits(return_X_y=True)
X_scaled = StandardScaler().fit_transform(X)

fig, axes = plt.subplots(3, 5, figsize=(12, 7))
component_counts = [1, 2, 5, 10, 30]

for col, nc in enumerate(component_counts):
    pca = PCA(n_components=nc)
    X_r = pca.inverse_transform(pca.fit_transform(X_scaled))
    # Un-standardise for display (approximate)
    axes[0, col].imshow(X[0].reshape(8, 8), cmap='gray')
    axes[0, col].set_title(f'Original' if col == 0 else '')
    axes[1, col].imshow(X_r[0].reshape(8, 8), cmap='gray')
    axes[1, col].set_title(f'n={nc}')

plt.tight_layout()
plt.show()

Reconstruction Error vs Number of Components

Plot reconstruction MSE against the number of components to see the information-loss curve. This is the quantitative version of the visual comparison. A sharp decrease in MSE as you add the first few components mirrors the scree plot, confirming that most information lives in a small subspace.

import matplotlib.pyplot as plt
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
from sklearn.datasets import load_digits
import numpy as np

X, _ = load_digits(return_X_y=True)
X_scaled = StandardScaler().fit_transform(X)

components = [1, 2, 5, 10, 20, 30, 40, 50, 64]
mse_values = []
for nc in components:
    pca = PCA(n_components=nc)
    X_r = pca.inverse_transform(pca.fit_transform(X_scaled))
    mse_values.append(np.mean((X_scaled - X_r) ** 2))

plt.plot(components, mse_values, marker='o')
plt.xlabel('Number of components')
plt.ylabel('Reconstruction MSE')
plt.title('Information Loss vs Compression')
plt.show()

Interpretting Reconstruction Error

At zero components, reconstruction error equals the total variance of the data. At full components, error is zero. The ratio 1 - explained_variance_ratio.sum() tells you the fraction of variance discarded. For most practical ML pipelines, keeping 95–99% of variance (and discarding 1–5%) loses very little predictive signal while significantly reducing feature count and training time.

from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
from sklearn.datasets import load_digits

X, _ = load_digits(return_X_y=True)
X_scaled = StandardScaler().fit_transform(X)

for nc in [5, 10, 20, 30, 40]:
    pca = PCA(n_components=nc)
    pca.fit(X_scaled)
    retained = pca.explained_variance_ratio_.sum()
    print(f'n_components={nc:2d}  retained={retained:.3f}  discarded={1-retained:.3f}')

Using inverse_transform in Practice

pca.inverse_transform(X_reduced) is a method on the fitted PCA object. It returns the data in the original feature space but with the information from discarded components zeroed out. This is useful for anomaly detection: reconstruct training data and flag points with high reconstruction error as outliers that the PCA model could not represent well.

from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
import numpy as np

# Simulated normal vs anomalous points
X_normal = np.random.randn(100, 10)
X_anomaly = np.random.randn(5, 10) * 10  # far from origin

X_all = np.vstack([X_normal, X_anomaly])
X_scaled = StandardScaler().fit_transform(X_all)

pca = PCA(n_components=5)
X_r = pca.inverse_transform(pca.fit_transform(X_scaled))
errors = np.mean((X_scaled - X_r) ** 2, axis=1)

print('Max error index:', np.argmax(errors), '(anomalies start at index 100)')

Whitening: Decorrelated Components with Unit Variance

Setting PCA(whiten=True) scales the projected scores so each component has unit variance. This removes correlations between components and can improve the performance of algorithms like SVMs or neural networks that are sensitive to feature scale. Whitening is standard preprocessing before training on PCA-reduced features.

from sklearn.decomposition import PCA
from sklearn.datasets import load_iris
from sklearn.preprocessing import StandardScaler
import numpy as np

X, _ = load_iris(return_X_y=True)
X_scaled = StandardScaler().fit_transform(X)

pca_white = PCA(n_components=3, whiten=True)
X_w = pca_white.fit_transform(X_scaled)

print('Component variances (should be 1.0):', np.var(X_w, axis=0).round(4))

PCA Limitations on Non-Linear Data

PCA finds only linear projections. If data lies on a curved surface — like a Swiss roll — PCA projects onto a flat plane, destroying the manifold structure. In such cases, consider Kernel PCA with an RBF kernel or non-linear alternatives like t-SNE or UMAP for exploration. For model preprocessing, however, linear PCA is usually sufficient and much faster.

Project and Reconstruct: Complete Workflow

A clean PCA pipeline always follows the same pattern: standardise, fit PCA on training data, transform train and test separately, optionally reconstruct to inspect quality.

from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
from sklearn.model_selection import train_test_split
from sklearn.datasets import load_digits
import numpy as np

X, y = load_digits(return_X_y=True)
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=0)

scaler = StandardScaler()
X_train_s = scaler.fit_transform(X_train)
X_test_s = scaler.transform(X_test)  # use train scaler

pca = PCA(n_components=0.95)
X_train_r = pca.fit_transform(X_train_s)   # fit only on train
X_test_r = pca.transform(X_test_s)         # transform test

print(f'Reduced: {X_train_r.shape[1]} components from 64 features')

Quick Check

Test your understanding of PCA projection and reconstruction from this lesson.

Lesson Recap

In this lesson you learned: pca.transform projects data into component space with shape (n_samples, n_components), pca.inverse_transform reconstructs data in original feature space with information from discarded components lost, and reconstruction error quantifies information loss and can flag anomalies. Next up we explore t-SNE — a non-linear technique for 2D visualisation of high-dimensional data.

Perguntas Frequentes

A aula “Projeção de dados e reconstrução a partir de componentes” é grátis?

Sim — o texto completo de “Projeção de dados e reconstrução a partir de componentes” é grátis para ler aqui na web. Para praticá-la interativamente (um editor de código integrado e um tutor de IA 24/7) e desbloquear o restante do curso de Machine Learning Academy, atualize para CoddyKit PRO. O curso de Machine Learning Academy inclui 4 aulas no total.

O que vou aprender em “Projeção de dados e reconstrução a partir de componentes”?

Os alunos transformarão um conjunto de dados para o espaço dos componentes principais, visualizarão a projeção 2D e reconstruirão as características originais para quantificar a perda de informação. Você pratica Machine Learning Academy com código prático que executa diretamente no navegador, e um tutor de IA 24/7 responde suas dúvidas enquanto trabalha na aula.

Preciso ter experiência prévia para começar Machine Learning Academy?

Nenhuma experiência prévia é necessária. Machine Learning Academy no CoddyKit é estruturado para alunos iniciantes até avançados, então você pode começar aqui ou desde o início e aprender no seu ritmo. Esta é a aula 2 de 4.

Quanto tempo leva a aula “Projeção de dados e reconstrução a partir de componentes”?

A maioria das aulas CoddyKit leva cerca de 5–10 minutos. Cada uma é compacta e interativa, então você faz progresso constante e retoma exatamente de onde parou entre web e app.

Posso escrever e executar código nesta aula de Machine Learning Academy?

Sim. Cada aula de Machine Learning Academy inclui um editor de código integrado, então você escreve e executa código real direto no navegador e recebe feedback de IA instantaneamente — nenhuma configuração local necessária.

Todas as aulas deste curso

  1. PCA: variância, autovetores e componentes principais
  2. Projeção de dados e reconstrução a partir de componentes
  3. t-SNE: preservação da vizinhança para visualização
  4. PCA como pré-processamento: velocidade e redução de ruído em pipelines
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