Impara PCA Intuition

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Definition

Principal component analysis (PCA) is a powerful technique that identifies new axes – called principal components – which are directions in your data that capture the most variance.

PCA keeps the directions where your data varies the most, as these capture the key patterns and structure.

Think of PCA like shining a flashlight on a 3D object and examining the shadow on a wall. The angle of the light changes the shadow's detail. PCA finds the best angle so the shadow, or projection, reveals the most about the object's shape. Similarly, PCA projects your data onto new axes to preserve as much variation as possible.


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import numpy as np
import matplotlib.pyplot as plt

# Generate a simple 2D dataset
np.random.seed(0)
mean = [0, 0]
cov = [[3, 2], [2, 2]]  # Covariance matrix
X = np.random.multivariate_normal(mean, cov, 200)

# Compute the mean of the data
mean_vector = np.mean(X, axis=0)

# Compute the covariance matrix and its eigenvectors
cov_matrix = np.cov(X.T)
eigenvalues, eigenvectors = np.linalg.eig(cov_matrix)

# First principal component (direction of maximum variance)
pc1 = eigenvectors[:, np.argmax(eigenvalues)]

# Plot the data
plt.figure(figsize=(8,6))
plt.scatter(X[:,0], X[:,1], alpha=0.3, label="Data points")
plt.quiver(
    mean_vector[0], mean_vector[1], 
    pc1[0], pc1[1], 
    angles='xy', scale_units='xy', scale=1.5, color='red', width=0.01,
    label="First principal component"
)
plt.xlabel("Feature 1")
plt.ylabel("Feature 2")
plt.title("Direction of Maximum Variance (First Principal Component)")
plt.legend()
plt.axis("equal")
plt.show()

By identifying the directions where your data varies the most, PCA allows you to reduce dimensions while preserving the most important information. Focusing on these directions of maximum variance ensures that the structure and patterns in your dataset remain clear. This understanding prepares you to explore the mathematical foundation of PCA in upcoming sections.

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Sezione 1. Capitolo 4

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Sezione 1. Capitolo 4