Visualizing Decorrelated Features
When you examine a dataset with correlated features, you often notice that the data points form an elongated cloud in scatter plots. This shape reflects the underlying relationships between variables. Whitening is a transformation that changes both the shape and orientation of this data cloud. After whitening, the features become uncorrelated and have unit variance, causing the cloud to appear more spherical and aligned with the axes. This transformation is crucial for algorithms sensitive to feature correlation, as it ensures each feature contributes independently.
123456789101112131415161718192021222324252627282930313233343536import numpy as np import matplotlib.pyplot as plt from sklearn.preprocessing import StandardScaler # Create a correlated 2D dataset np.random.seed(0) mean = [0, 0] cov = [[3, 2.5], [2.5, 3]] # strong positive correlation X = np.random.multivariate_normal(mean, cov, 500) # Standardize the data scaler = StandardScaler() X_std = scaler.fit_transform(X) # Whitening transformation using eigenvalue decomposition cov_matrix = np.cov(X_std, rowvar=False) eigvals, eigvecs = np.linalg.eigh(cov_matrix) whitening_matrix = eigvecs @ np.diag(1.0 / np.sqrt(eigvals)) @ eigvecs.T X_white = X_std @ whitening_matrix # Plot before and after whitening fig, axes = plt.subplots(1, 2, figsize=(12, 5)) axes[0].scatter(X_std[:, 0], X_std[:, 1], alpha=0.5) axes[0].set_title("Before Whitening") axes[0].set_xlabel("Feature 1") axes[0].set_ylabel("Feature 2") axes[0].axis("equal") axes[1].scatter(X_white[:, 0], X_white[:, 1], alpha=0.5, color="green") axes[1].set_title("After Whitening") axes[1].set_xlabel("Feature 1") axes[1].set_ylabel("Feature 2") axes[1].axis("equal") plt.tight_layout() plt.show()
Sphering is another term for whitening. It refers to transforming a dataset so that its covariance matrix becomes the identity matrix, making the data have zero correlation and unit variance along all axes.
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When you examine a dataset with correlated features, you often notice that the data points form an elongated cloud in scatter plots. This shape reflects the underlying relationships between variables. Whitening is a transformation that changes both the shape and orientation of this data cloud. After whitening, the features become uncorrelated and have unit variance, causing the cloud to appear more spherical and aligned with the axes. This transformation is crucial for algorithms sensitive to feature correlation, as it ensures each feature contributes independently.
123456789101112131415161718192021222324252627282930313233343536import numpy as np import matplotlib.pyplot as plt from sklearn.preprocessing import StandardScaler # Create a correlated 2D dataset np.random.seed(0) mean = [0, 0] cov = [[3, 2.5], [2.5, 3]] # strong positive correlation X = np.random.multivariate_normal(mean, cov, 500) # Standardize the data scaler = StandardScaler() X_std = scaler.fit_transform(X) # Whitening transformation using eigenvalue decomposition cov_matrix = np.cov(X_std, rowvar=False) eigvals, eigvecs = np.linalg.eigh(cov_matrix) whitening_matrix = eigvecs @ np.diag(1.0 / np.sqrt(eigvals)) @ eigvecs.T X_white = X_std @ whitening_matrix # Plot before and after whitening fig, axes = plt.subplots(1, 2, figsize=(12, 5)) axes[0].scatter(X_std[:, 0], X_std[:, 1], alpha=0.5) axes[0].set_title("Before Whitening") axes[0].set_xlabel("Feature 1") axes[0].set_ylabel("Feature 2") axes[0].axis("equal") axes[1].scatter(X_white[:, 0], X_white[:, 1], alpha=0.5, color="green") axes[1].set_title("After Whitening") axes[1].set_xlabel("Feature 1") axes[1].set_ylabel("Feature 2") axes[1].axis("equal") plt.tight_layout() plt.show()
Sphering is another term for whitening. It refers to transforming a dataset so that its covariance matrix becomes the identity matrix, making the data have zero correlation and unit variance along all axes.
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