Teaching and Lecture Recordings
Computational Statistics and Data Analysis
This lecture introduces core concepts in probability, statistical inference, and computational modeling, with a progression from classical methods to modern machine learning approaches for data analysis.
Outline
- Overview and basics of probability theory
- Discrete/continuous distributions and exponential-family models
- Moment generating functions and statistical models
- Statistical inference and parameter estimation
- Numerical methods and statistical hypothesis tests
- Asymptotic tests, CLT, and bootstrap-based tests
- Linear and nonlinear regression models
- Neural networks and nonlinear regression
- Classification models I and II
- Regularization and dimensionality reduction
- Latent variable models
- Variational inference and generative adversarial networks
Dynamical Systems Theory in Machine Learning
Dynamical systems theory provides a mathematical framework for understanding processes that evolve in time, from neural activity to engineered systems. The course connects classical DST concepts to modern ML-based system reconstruction.
Outline
- Introduction to dynamical systems and linear ODE systems
- Equilibrium/fixed-point properties and stability analysis
- Nonlinear ODE systems, limit cycles, and phase locking
- Hamiltonian systems, potentials/gradients, and Lyapunov functions
- Recursive maps, Poincare maps, and symbolic dynamics
- Chaos, fractals, invariant measures, and Lyapunov exponents
- Local and global bifurcations
- Dynamical systems from data: delay embedding and dimension estimation
- ML models for DS reconstruction: SINDy, reservoir computing, and RNNs
- RNN training dynamics: exploding gradients and bifurcation effects
- Neural ODEs, PINNs, and neural operators
- Manifold detection, multimodal training, and tracking across tipping points
- Out-of-domain generalization and foundation models for DS reconstruction
Time Series Models: From Statistics to AI
Time series are central across science, medicine, economics, and engineering. This course bridges statistical models and modern AI systems for forecasting and mechanistic understanding of latent dynamics.
Outline
- Statistical foundations and model classes for time series
- Linear ARMA and multivariate AR models
- Granger causality and non-Gaussian count/point-process models
- Linear dynamical systems and latent-variable state-space models
- EM, Kalman filtering/smoothing, and non-Gaussian LDS inference
- Intro to nonlinear dynamics for time-series modeling
- Recurrent neural networks and gradient-based training (BPTT/RTRL)
- Long-term dependencies, LSTM/GRU, and generalized teacher forcing
- Modern recurrent architectures: Reservoir Computing, AL-RNN, and Mamba
- Stochastic/multimodal RNNs, VAEs, and variational inference
- Neural ODEs for continuous-time point processes
- Attention/self-attention, transformers, and TS foundation models
- In-context learning and zero-shot inference for TS models