In this tutorial, we will delve into the basics of Principal Component Analysis (PCA), a dimensionality reduction technique used in machine learning. PCA is widely used for data compression and feature extraction.

What is PCA?

PCA is a statistical procedure that uses orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components.

Key Points of PCA:

  • Dimensionality Reduction: PCA reduces the dimensionality of the data, making it easier to visualize and analyze.
  • Feature Extraction: PCA extracts the most significant features from the data.
  • Data Compression: PCA can be used to compress the data without losing much information.

PCA Workflow

  1. Standardize the Data: The first step in PCA is to standardize the data. This means that we subtract the mean and divide by the standard deviation for each feature.
  2. Compute the Covariance Matrix: The next step is to compute the covariance matrix of the standardized data.
  3. Compute Eigenvectors and Eigenvalues: Eigenvectors and eigenvalues are computed from the covariance matrix.
  4. Sort Eigenvectors by Eigenvalues: The eigenvectors are sorted in descending order of their corresponding eigenvalues.
  5. Project Data onto Principal Components: Finally, the data is projected onto the first few principal components.

Example

Let's say we have a dataset with three features: X, Y, and Z. We want to reduce the dimensionality of this dataset using PCA.

import numpy as np

# Generate some random data
data = np.random.randn(100, 3)

# Standardize the data
mean = np.mean(data, axis=0)
std = np.std(data, axis=0)
standardized_data = (data - mean) / std

# Compute the covariance matrix
cov_matrix = np.cov(standardized_data, rowvar=False)

# Compute eigenvectors and eigenvalues
eigenvalues, eigenvectors = np.linalg.eigh(cov_matrix)

# Sort eigenvectors by eigenvalues
sorted_indices = np.argsort(eigenvalues)[::-1]
sorted_eigenvectors = eigenvectors[:, sorted_indices]

# Project data onto the first two principal components
pca_data = np.dot(standardized_data, sorted_eigenvectors[:, :2])

Further Reading

For more information on PCA, you can refer to the following resources:

[center] PCA [center]