If you're new to PCA and want to understand the basics, this tutorial is for you! PCA is a powerful technique used in data analysis and machine learning to reduce the dimensionality of large datasets while retaining trends and patterns. Let's dive in!

What is PCA?

PCA is a statistical method that converts a large set of variables into a smaller one that still contains most of the information in the large set. Essentially, it's a way to compress data without losing too much information.

PCA Applications

  • Feature extraction: Reducing the number of variables for a dataset while preserving the most important information.
  • Visualization: Creating 2D or 3D visualizations of high-dimensional data.
  • Noise reduction: Reducing noise and redundancy in the data.

How PCA Works

  1. Standardize the data: Ensure that each feature has a mean of zero and a standard deviation of one.
  2. Compute the covariance matrix: Measure the relationship between variables.
  3. Calculate the eigenvectors and eigenvalues: These represent the directions and magnitudes of the new feature space.
  4. Choose the top k eigenvectors: Based on the eigenvalues, select the eigenvectors that capture the most variance.
  5. Transform the data: Use the selected eigenvectors to transform the original data into the reduced feature space.

Example

Suppose we have a dataset with 3 features: height, weight, and age. PCA can help us understand how these features are related and potentially reduce the dimensionality to just 2 features.

Data Standardization

Before applying PCA, it's important to standardize the data:

height: [170, 175, 180]
weight: [60, 65, 70]
age: [25, 30, 35]

Standardized data:
height: [0.447, 0.527, 0.606]
weight: [0.000, 0.000, 0.000]
age: [-1.224, -1.092, -0.950]

Covariance Matrix

The covariance matrix will give us insights into the relationships between the features.

Covariance Matrix:
|   | height | weight | age |
|---|--------|--------|-----|
| height |   0.014 |  0.005 |  0.002 |
| weight |  0.005 |   0.022 |  0.009 |
| age   |  0.002 |   0.009 |   0.019 |

Eigenvectors and Eigenvalues

Next, we calculate the eigenvectors and eigenvalues of the covariance matrix. The eigenvectors represent the directions of the new feature space, while the eigenvalues represent the magnitude.

Eigenvectors and Eigenvalues:
Eigenvector 1: [0.765, 0.423, 0.518]
Eigenvalue 1:  0.032

Eigenvector 2: [-0.423, 0.765, 0.518]
Eigenvalue 2:  0.014

Eigenvector 3: [-0.518, -0.518, 0.000]
Eigenvalue 3:  0.002

Choose Top k Eigenvectors

For our example, we'll choose the first two eigenvectors because they capture the most variance.

Selected Eigenvectors:
Eigenvector 1: [0.765, 0.423, 0.518]
Eigenvector 2: [-0.423, 0.765, 0.518]

Transform the Data

Finally, we transform the original data using the selected eigenvectors.

Transformed Data:
[0.617, 0.411]
[0.648, 0.395]
[0.679, 0.388]

Learn More

If you're interested in diving deeper into PCA or learning more about data analysis and machine learning, check out our Introduction to Machine Learning tutorial.

PCA Illustration