Week 2 Summary: Physics of data formation

Cross-Book Summary

1. Signal Formation & Noise

• Measurement Mapping: Signal = physical state convolved with instrument PSF.
• Stochastics: Data is a stochastic process.
• Noise as Prior: Noise distributions reveal detector physics.

2. Dimension Reduction & Structure

• Dimensionality Curse: High dimension, low intrinsic rank.
• SVD vs. PCA: PCA uses covariance; SVD factorizes matrix efficiently.
• Principal Variations: Top singular vectors capture main physical phenomena.
• Reduced Order Modeling: PCA scores map to processing parameters.

90-Minute Lecture Strategy

Part 1: Measurement Chain

• Signal formation process.
• Resolution and Contrast.
• Nyquist-Shannon sampling limit.

Part 2: Statistical Foundations

• Variance, Covariance, Correlation.
• Covariance Matrix geometry.
• Noise distributions.

Part 3: Dimension Reduction

• Finding natural coordinates.
• PCA steps: Center, Covariance, Eigen-decomposition.
• SVD mechanics.
• Scree Plots and explained variance.

Part 4: Materials Case Studies

• Spectral Denoising with PCA.
• Time-series Compression.
• Microstructure Eigen-modes.

Part 5: Physics-ML Integration

• PCA as physical mode finder.
• Non-linear limits of PCA.
• Fourier periodicity exercise.

Quarto Website Update (Summary)

Summary for ML-PC Week 2:
- Connects physical data acquisition to mathematical representation. - Analyzes signal formation, resolution, and noise as priors. - Introduces PCA and SVD for low-dimensional structure discovery. - Demonstrates dimensionality reduction in high-dimensional datasets.