Principal Component Analysis (PCA) is a dimensionality reduction technique that transforms data to a new coordinate system where the axes (principal components) are ordered by the amount of variance they explain.
The PCA Algorithm:
- Standardization: Center the data by subtracting the mean, and optionally scale it to unit variance
- Covariance Matrix: Compute the covariance matrix of the standardized data
- Eigendecomposition: Find the eigenvectors and eigenvalues of the covariance matrix
- Sort Components: Order the eigenvectors by their corresponding eigenvalues (highest to lowest)
- Project Data: Transform the original data onto the new coordinate system defined by the principal components
Key Properties:
- Principal Components: Directions in which the data varies the most
- Eigenvalues: Represent the amount of variance explained by each principal component
- Variance Explained: Percentage of total variation captured by each principal component
- Orthogonality: Principal components are perpendicular to each other
Applications:
- Visualization of high-dimensional data
- Noise reduction and data preprocessing
- Feature extraction and selection
- Compression of high-dimensional data
Limitations:
- Only captures linear relationships in the data
- Sensitive to the relative scaling of the original variables
- May not work well if the data has non-linear structures
- Interpretation of principal components can be difficult