Understanding the Curse of Dimensionality
In statistics, more data often means more insights. However, more variables (dimensions) create exponential complexity:
- 2 variables: Easy to visualize on a 2D plot
- 3 variables: Still manageable in 3D space
- 10+ variables: Impossible to visualize, computationally expensive
This is the curse of dimensionality. To combat it, we use dimensionality reduction techniques like Principal Component Analysis (PCA).
Spectral Theory: Finding Order in Chaos
Spectral theory helps us:
- Identify the most important dimensions in our data
- Reduce complexity while preserving information
- Transform correlated variables into independent components
These techniques fall under spectral analysis - mathematical methods that reveal hidden structure in complex data.
Interactive 3D Visualization
This demo generates samples from a multivariate normal distribution. You can see how data distributes in 3D space and how variables correlate with each other.
How It Works
- Random Sampling: Generate points using the Mersenne Twister algorithm
- Normal Distribution: Convert uniform random numbers to normal distribution
- Correlation: Apply covariance matrix using Cholesky decomposition
- Visualization: Render in 3D space with WebGL
Using the Covariance Matrix
The covariance matrix controls how variables relate. Each row represents how one variable depends on the others:
[1.0 0.5 0.0]
[0.5 1.0 0.3]
[0.0 0.3 1.0]This means:
- Variables 1 and 2 are moderately correlated (0.5)
- Variables 2 and 3 are weakly correlated (0.3)
- Variables 1 and 3 are independent (0.0)
Controls
- Left-click + drag: Rotate view around axis center
- Right-click + drag: Pan camera position
- Scroll: Zoom in/out
- Covariance matrix: Adjust correlations between variables
What You’re Seeing
Each point represents a sample with three variables (x, y, z). The shape of the point cloud reveals:
- Elongation: Indicates correlation
- Orientation: Shows which variables correlate
- Density: Higher in the center (normal distribution)
Applications
This type of visualization helps understand:
- PCA: Find principal components (eigenvectors of covariance matrix)
- Data exploration: Identify patterns and outliers
- Machine learning: Understand feature relationships
- Simulation: Test statistical methods
Technical Details
The implementation uses:
- Mersenne Twister: High-quality random number generator
- Box-Muller transform: Converts uniform to normal distribution
- Cholesky decomposition: Efficiently applies covariance structure
- WebGL: Hardware-accelerated 3D rendering
This runs entirely in your browser with no server required. Performance remains excellent even with thousands of points.
Try It Yourself
Experiment with different covariance matrices:
High correlation (elongated cloud):
[1.0 0.9 0.9]
[0.9 1.0 0.9]
[0.9 0.9 1.0]Independent variables (spherical cloud):
[1.0 0.0 0.0]
[0.0 1.0 0.0]
[0.0 0.0 1.0]Mixed correlations:
[1.0 0.8 0.0]
[0.8 1.0 0.0]
[0.0 0.0 1.0]Observe how the point cloud’s shape changes with each configuration!