Visualizing Normalized Vectors
Understanding how normalization transforms data in two-dimensional space is essential for grasping its impact on machine learning workflows. When you normalize a set of 2D points, you change not only their scale but often their orientation relative to the origin. Suppose you have several points scattered across a plane — each with its own magnitude (distance from the origin) and direction. Applying normalization, such as L2 normalization, adjusts each point so that it lies on a unit circle centered at the origin. This operation preserves the direction of each vector from the origin but forces all points to have the same magnitude, making them directly comparable regardless of their original scale. Such a transformation is particularly useful when the absolute scale of vectors is irrelevant, but their directions encode meaningful information.
1234567891011121314151617181920212223242526272829import numpy as np import matplotlib.pyplot as plt # Original 2D points points = np.array([ [3, 4], [1, 7], [5, 2], [6, 9], [8, 1] ]) # Compute L2 norms for each point norms = np.linalg.norm(points, axis=1, keepdims=True) # L2-normalize each point normalized_points = points / norms # Plotting plt.figure(figsize=(7, 7)) plt.scatter(points[:, 0], points[:, 1], color='blue', label='Original Points') plt.scatter(normalized_points[:, 0], normalized_points[:, 1], color='red', label='Normalized Points') plt.xlabel('X') plt.ylabel('Y') plt.title('Original vs. L2-Normalized 2D Points') plt.legend() plt.grid(True) plt.axis('equal') plt.show()
Distance-based algorithms, such as k-nearest neighbors, rely heavily on how distances are measured between data points. If features are on different scales, the algorithm may be biased toward features with larger ranges, distorting similarity assessments. Normalizing vectors ensures that each point contributes equally to distance calculations, making the analysis fair and meaningful regardless of the original feature scales.
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Understanding how normalization transforms data in two-dimensional space is essential for grasping its impact on machine learning workflows. When you normalize a set of 2D points, you change not only their scale but often their orientation relative to the origin. Suppose you have several points scattered across a plane — each with its own magnitude (distance from the origin) and direction. Applying normalization, such as L2 normalization, adjusts each point so that it lies on a unit circle centered at the origin. This operation preserves the direction of each vector from the origin but forces all points to have the same magnitude, making them directly comparable regardless of their original scale. Such a transformation is particularly useful when the absolute scale of vectors is irrelevant, but their directions encode meaningful information.
1234567891011121314151617181920212223242526272829import numpy as np import matplotlib.pyplot as plt # Original 2D points points = np.array([ [3, 4], [1, 7], [5, 2], [6, 9], [8, 1] ]) # Compute L2 norms for each point norms = np.linalg.norm(points, axis=1, keepdims=True) # L2-normalize each point normalized_points = points / norms # Plotting plt.figure(figsize=(7, 7)) plt.scatter(points[:, 0], points[:, 1], color='blue', label='Original Points') plt.scatter(normalized_points[:, 0], normalized_points[:, 1], color='red', label='Normalized Points') plt.xlabel('X') plt.ylabel('Y') plt.title('Original vs. L2-Normalized 2D Points') plt.legend() plt.grid(True) plt.axis('equal') plt.show()
Distance-based algorithms, such as k-nearest neighbors, rely heavily on how distances are measured between data points. If features are on different scales, the algorithm may be biased toward features with larger ranges, distorting similarity assessments. Normalizing vectors ensures that each point contributes equally to distance calculations, making the analysis fair and meaningful regardless of the original feature scales.
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