Theory Overview

This section provides the mathematical foundations for the algorithms implemented in sppcax. The primary model is Dynamic Factor Analysis (DFA), a Linear Gaussian State Space Model with ARD priors for automatic sparsification. Factor Analysis (FA) and Probabilistic PCA are obtained as special cases when there are no temporal dynamics.

• Bayesian Dynamic Factor Analysis
  • Introduction
  • Model Definition
  • Prior Distributions
  • Variational Inference
  • Factor Analysis and PCA as Special Cases
  • Training Pipeline
  • References
• Inference Methods: EM, VBEM, and Gibbs
  • Introduction
  • Shared M-Step Pipeline
  • EM (Maximum A Posteriori)
  • VBEM (Variational Bayes EM)
  • Blocked Gibbs Sampling
  • Algorithm Summaries
  • See Also
  • References
• Factor Analysis and PCA
  • Introduction
  • Model Definition
  • Prior Distributions
  • Variational Inference
  • Evidence Lower Bound (ELBO)
  • Update Equations
  • Handling Missing Data
  • References
• Parameter Expansion for Dynamic Factor Analysis
  • Introduction
  • Parameter Expanded VBEM (PX-VBEM)
  • See Also
  • References
• Bayesian Model Reduction
  • Introduction
  • Principle of Bayesian Model Reduction
  • Mathematical Formulation
  • Variational Approximation
  • Computing \(\Delta F\)
  • Gibbs Sampling with Indian Buffet Process Prior
  • Posterior Correction After Pruning
  • See Also
  • References
• Examples
  • Testing PX-VBEM for Bayesian Factor Analysis
  • Testing PX-VBEM for Dynamic Factor Analysis
  • Testing Observation Masking
  • Part 1: Factor Analysis (Static Mode)
  • Part 2: Dynamic Factor Analysis (DFA)